research article integrated guidance and control method

Research ArticleIntegrated Guidance and Control Method forthe Interception of Maneuvering Hypersonic VehicleBased on High Order Sliding Mode Approach

Kang Chen1 Bin Fu1 Yuening Ding2 and Jie Yan1

1College of Astronautics Northwestern Polytechnical University Xirsquoan 710072 China2Flight Automation Control Research Institute Xirsquoan 710065 China

Correspondence should be addressed to Bin Fu binfumailnwpueducn

Received 27 April 2015 Revised 26 July 2015 Accepted 29 July 2015

Academic Editor Giuseppe Rega

Copyright copy 2015 Kang Chen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper focuses on the integrated guidance and control (IGC) method applied in the interception of maneuvering near spacehypersonic vehicles using the homogeneous high order sliding mode (HOSM) approach The IGC model is derived by combiningthe target-missile relative motion and dynamic equations Then a fourth-order sliding mode controller is implemented in theaugmented IGC model To estimate the high order derivatives of the sliding manifold which is required in the HOSMmethod anArbitrary Order Robust Exact Differentiator is presented At last the idea of virtual control is introduced to alleviate the chatteringof the control input without using any saturation functions which may lead to a loss of the robustness And the stability of theclosed-loop system with presented fourth-order homogeneous HOSM controller is also proved theoretically Finally simulationresults are provided and analyzed to demonstrate the effectiveness of the proposed method in three typical engagement scenarios

1 Introduction

Because of its speed advantage and flexible maneuverabilitythe hypersonic vehicle may certainly become a severe threatin the future battlefield to deal with the threat the researchon the interception of the hypersonic vehicle is already onthe agenda Primarily the interception of the hypersonicvehicle faces the following issues (1) when the speed ofits target is much faster than the interception missile theeffective attack area of the traditional guidance law greatlyshrinks and it is impossible to accomplish the tail-chase orbackward interception (2) at the high altitude of 25 to 40 kmwhere the hypersonic vehicle flies the air is relatively thinthe aerodynamic efficiency of an interception missile is lowand there is a limited usable overload for the interceptor(3) it is difficult to destroy such a hypersonic target withthe traditional destructive means such as near explosionfragments requiring that the interception missile should useknock-on collision as much as possible to attack the targetnamely minimal target missing As the requirements for

guidance accuracy are higher researchers do massive workto advance the guidance and control theory

During the past decades the proportional navigation(PN) guidance law is a popular and widely used method inmissile interception missions for its ease of implementationand high efficiency The principle of PN guidance law in [1]is that the commanded normal acceleration of the missileis proportional to the line-of-sight (LOS) rate which issimple and effective under a wide range of engagementscenarios However the use of the PN guidance law is alsolimited for example as the distance between the missileand its target is closing or the target acts an unpredictablemaneuver the LOS rate may grow extremely fast andsubsequently the overload needed at the end phase alsodiverges at last As for the overload autopilot accountingfor a second-order dynamic the real acceleration responseof the missile to the fast-changing high-frequency overloadcommand lags behind and attenuates and eventually caused alarge miss distance Therefore many new guidance laws cropup

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 648231 19 pageshttpdxdoiorg1011552015648231

2 Mathematical Problems in Engineering

Being famous for its good robustness to bounded distur-bances the sliding mode control method is introduced intothe study of guidance law In [2] Zhou et al gave the con-ditions for the sliding mode motion of a linear time-varyingsystem not to be disturbed by the disturbances and param-eter perturbations and proposed an adaptive sliding modeguidance law (ASMG) simulation results demonstrated thatthe ASMG is robust to uncertainties like target accelerationTo get finite time convergence modified SMC like terminalsliding mode control (TSMC) is proposed In [3] Zeng andHu combine the advantages of linear and terminal slidingmode controls which guarantee the convergence of trackingerrors in finite time And then nonsingular terminal slidingmode control (NTSMC) is introduced in the guidance lawdesign In [4] Kumar et al proposed a nonsingular terminalsliding mode guidance law with finite time convergencewhich avoids the singularity that may lead to the saturationof the control Integral SMC (ISMC) like that introduced in[5] is another revised sliding mode control which introducedthe integral sliding mode scheme to the guidance law designthe proposed ISMC guidance law provided a smaller controlmagnitude than the traditional sliding mode design Theadvance of the 119867

infincontroller greatly enriched the methods

to implement a robust control system In [6] Savkin etal suitably modified the 119867

infincontrol theory and provided

an effective framework for the precision missile guidanceproblem which showed much better performance than thelinear quadratic optimal guidance law in the computersimulation In [7] Chen and Yang proposed a mixed 119867

2119867infin

guidance design against maneuvering targets in which thecomplete nonlinear kinematics of the pursuit-evasionmotionwas considered

In addition to the separated guidance and control lawthe integrated guidance and control method is under heateddiscussion According to the relative motion between thetarget and the missile the traditional guidance law calculatesthe overload needed to hit the target and inputs the overloadcommand into the overload autopilot while the integratedguidance and control (IGC) method gives the rudder deflec-tion command directly to the missile according to therelative motion which evidently responds more quickly In[8] Menon et al designed an IGCmethod by using the linearquadratic optimal theory but the robustness of the system isnot satisfactory In [9] Vaddi et al provided a fully numericalmethodology for deriving state-dependent Riccati equationcontrollers for arbitrarily complex dynamic systems andapplied it in the IGC design of a missile Simulation resultsdemonstrated the effectiveness of the method In [10] Xinet al employed the theta-D method to give an approximateclosed-form suboptimal feedback controller to the nonlinearinfinite-horizon IGC problem Taking another look slidingmode approaches are also employed in the IGC design ofhoming missiles In [11] Shima et al proposed the slidingmode integrated guidance and control method based on theZEM Shtessel and Tournes doesmassive research on the highorder sliding mode controller design and proposed his newmethod In [12] he designed the high order sliding modeguidance law based on the smooth second-order slidingmode control (SSOSMC) which is smoother in high orders

than the traditional second-order sliding mode guidance lawIn [13] based on the geometrical homogeneity theory Donget al designed the tranquility control law for the integratedguidance and control model which however has rathermore parameters and is sensitive to parameters and theparameters must be carefully selected In [14] Mingzhe andGuangren used the sliding mode control theory to design theadaptive nonlinear feedback controller which is primarily forfixed ground target being unable to deal with the vehementperturbation caused by target maneuvering

Motivated by the aforementioned considerations thiswork will design an IGC scheme for the interception of thenear space maneuvering hypersonic vehicles Firstly a line-of-sight (LOS) rate feedback scheme is adopted to derive theIGC law and as the relative order of the control input to theLOS rate is higher than one a HOSM approach is introducedSecondly to implement the HOSM approach the 119899 timesderivations of the sliding manifold 119878 119878

119878 119878

(119899) must beknown However reconstruction of each 119878 119878

119878 119878

(119899) bytheir analytical expression could be rather difficult in prac-tice An alternative way adopted in this paper is to use theArbitrary-Order Robust Exact Differentiator (AORED) toestimate the derivations And next to alleviate the chatteringphenomenon caused by the HOSM controller the idea ofvirtual control is introducedThe virtual control V is designedand used as the control input of a system extended from theoriginal one and the real control 119906 acting on the real systemis obtained by integrating the virtual control V Benefittingfrom the integration element the real control input 119906 couldbe smooth enough for the implementation without reducingthe robustness of the HOSM method Finally the proposedmethod is implemented in a 3-dof model

The remaining part of this paper is organized as followsIn Section 2 the IGC model is derived in the longitudinalplane In Section 3 the quasi-continuous HOSM controlleras well as the AORED is designed In Section 4 the baselineseparated guidance laws and controller are given In Sec-tion 5 the numerical simulations are demonstrated in threetypical engagement scenarios And conclusions are made inSection 6

2 Integrated Guidance and Control Model

The traditional guidance and control algorithm usually usesthe guidance loop as its outer loop and is only responsible forgiving commanded overload then the control loop is onlyresponsible for tracking the overload command eventuallyachieving the missilersquos guidance toward its target Althoughit is always desirable to design the control loop or theautopilot to have better dynamic performance in actualitythe controller always has some delay and attenuation Asa result the missile always has some error in acting theoverload command That is one of the reasons why aninterceptor misses its target

The integrated guidance and control algorithm combinesthe guidance and control loops into one loop and avoidsthe delay and attenuation caused by them Its architecture isshown in Figure 1

Mathematical Problems in Engineering 3

Seeker

Guidancefiltering

Integratedguidance and control

ActuatorMissile

airframe

Missile position

Target position

Missile acceleration

Target acceleration

Guidancelaw

Autopilot(controller) Separated guidance and control

Integrated guidance and control +minus

120575CnC

q

120575120575C

q

q nm

Figure 1 Integrated guidance and control

O

Y1

X1

M

aM

xb

120572120579M

q

VM

T

VT

120579T

aT

120599

Figure 2 Engagement geometry

21 The Engagement Dynamics Without loss of generalitywe present hereinafter only the subsystems that govern themotion of an interceptor in its longitudinal plane The planarengagement between the interceptor and its target is shownin Figure 2

The study of guidance laws usually regards themissile andits target as mass points deliberates on the mass point of itsrigid motion and ignores the attitude of the missile airframeand changes in its attitude By contrast the integrated guid-ance and control method takes the dynamic characteristicsof the missile airframe into consideration therefore thederivation involves the pitch angle pitch rate and angle ofattack of the missile as shown in Figure 2

The motion equations of the missile in the longitudinalplane are as follows

119889119881119872

119889119905=

1

119898(119875 cos120572 minus 119883 minus 119898119892 sin 120579

119872) (1)

119889120579119872

119889119905=

1

119898119881119872

(119875 sin120572 + 119884 minus 119898119892 cos 120579119872

) (2)

119889119909119872

119889119905= 119881119872cos 120579119872

(3)

119889119910119872

119889119905= 119881119872sin 120579119872

(4)

119889120596119885

119889119905=

119872119885

119869119885

(5)

119889120599

119889119905= 120596119885 (6)

120572 = 120599 minus 120579 (7)

where (119909119872

119910119872

) is the position 119898 is the mass 119883 is the axialforce 119875 is the thrust force the missilersquos velocity is 119881

119872 and its

flight path angle is 120579119872 the LOS angle between themissile and

its target is 119902 the relative distance between the missile and itstarget is 119903 the velocity of the target is 119881

119879 and its flight path

angle is 120579119879 119909119887is the axle of the missilersquos airframe the pitch

angle is 120599 the angle of attack is 120572The motion equations of the target are as follows

119889119909119879

119889119905= minus119881119879cos 120579119879

119889119910119879

119889119905= 119881119879sin 120579119879

119889120579119879

119889119905=

119886119879119873

119881119879

(8)

The interception is characterized by two variablesnamely the target range and the LOS angle The kinematicequations are expressed by the following relations

119889119903

119889119905= minus119881119872cos (120579

119872minus 119902) minus 119881

119879cos (120579

119879+ 119902) (9)

119903119889119902

119889119905= minus119881119872sin (120579119872

minus 119902) + 119881119879sin (120579119879

+ 119902) (10)

22 The Model Simplification To simplify the model andmake further derivations wemake the following two assump-tions

Assumption 1 Within the terminal phase of the interceptionthe missile has no thrust and its gravity is not taken intoaccount

Assumption 2 Within the terminal phase of the interceptionthe missilersquos speed does not change


Then (1) and (2) can be reformed as below

119889119881119872

119889119905= 0

119889120579119872

119889119905=

119884

119898119881119872

(11)

Denote that 119886119872119873

= 119884119898 119884 is the normal force 119886119872119873

119886119879119873

are the normal acceleration of the missile and targetrespectively and then

120579119872

=119886119872119873

119881119872

120579119879

=119886119879119873

119881119879

(12)

The normal force 119884 and the pitch moment 119872119885acting on

the missile are usually expressed respectively as follows

119884 = 119862120572

119884120572119876119878ref + 119862

120575119885

119884120575119885119876119878ref (13)

119872119885

= 119898120575119885

119885120575119885119876119878ref119897 + 119898

120572

119885120572119876119878ref119897 (14)

where119862120572

119884is the coefficient of normal force caused by the angle

of attack 119862120575119885

119884is the coefficient of normal force caused by the

rudder deflection angle 120575119885 119876 is the dynamic pressure 119878ref is

the reference area 119897 is the reference length But the normalforce produced by the rudder deflection angle 120575

119885is orders of

magnitude smaller than that produced by the angle of attackso (13) is simplified as

119884 = 119862120572

119884120572119876119878ref (15)

Then

119886119872119873

=119884

119898=

119862120572

119884120572119876119878ref

119898 (16)

Then the dynamic equations can be simplified as

119889120579119872

119889119905=

119862120572

119884120572119876119878ref

119898119881119872

(17)

119889120596119885

119889119905=

119898120575119885

119885120575119885

119876119878ref119897 + 119898120572

119885120572119876119878ref119897

119869119885

(18)

119889120599

119889119905= 120596119885 (19)

120572 = 120599 minus 120579119872

(20)

23 The Integrated Guidance and Control Model Follow-ing the above derivation we select the state variables as

( 119902 120579119872

120572 120596119885)119879 and obtain the following nonlinear integrated

guidance and control model

119902 = minus2119903

119903sdot 119902 minus

119862120572

119884119876119878ref

119898119903cos (120579

119872minus 119902) sdot 120572

+119886119879119873

119903cos (120579

119879+ 119902)

119903 119902 = minus119881119872sin (120579119872

minus 119902) + 119881119879sin (120579119879

+ 119902)

120579119872

=1

119898119881119872

119862120572

119884119876119878ref sdot 120572

= 120596119885

minus1

119898119881119872

119862120572

119884119876119878ref sdot 120572

119885

=119898120575119885

119885119876119878ref119897

119869119885

sdot 120575119885

+119898120572

119885119876119878ref119897

119869119885

120572

(21)

24 The Relative Degree of Control Input To obtain therelative degree of the control input 120575 of the integratedguidance and control method we keep on deriving the LOSangular velocity 119902 until the explicit formula of its derivativeof a certain order contains the control input

The derivation of (10) produces119902

=1

119903[minus2 119903 119902 minus 119886

119872119873cos (120579

119872minus 119902) + 119886

119879119873cos (120579

119879+ 119902)]

(22)

The LOS angular velocity is expressed as the derivativeof the first order and does not contain the control input 120575

explicitly The continuous derivation of the above equationproduces

119902 =

1

119903minus2 119903 119902 minus 3 119903 119902 minus [119886

119872119873( 119902 minus 120579

119872) sin (120579

119872minus 119902)

+ 119886119872119873

cos (120579119872

minus 119902)] minus [119886119879119873

( 119902 + 120579119879) sin (120579

119879+ 119902)

minus 119886119879119873

cos (120579119879

+ 119902)]

(23)

where the LOS angular rate is expressed as the differentiatingof the second order and 119886

119872can be expressed as

119886119872119873

=119862120572

119884119876119878ref

119898 =

119862120572

119884119876119878ref

119898( 120599 minus 120579

119872)

=119862120572

119884119876119878ref

119898(120596119885

minus 120579119872

)

(24)

Although the control volume 120575119885does not appear in the

derivative of the second order (18) shows that 119885contains

120575119885 We continue to derive (23) and substitute 119886

119872as follows

119886119872119873

=119862120572

119884119876119878ref

119898(119885

minus119886119872

119881119872

)

=119862120572

119884119876119878ref

119898

119898120575119885

119885119876119878ref119897ref

119869119885

sdot 120575119885

+119862120572

119884119876119878ref

119898(

119898120572

119885119876119878ref119897ref

119869119885

120572 minus119886119872

119881119872

)

(25)


Thus

119902(4)

= 119891120575119885

sdot 120575119885

+1

119903(1198911

+ 1198912

+ 1198913

+ 1198914

+ 1198915

+ 1198916) (26)

where

119891120575119885

= minus1

119903

119862120572

119884119876119878ref cos 120578

119872

119898

119898120575119885

119885119876119878ref119897ref

119869119885

(27)

1198911

= (120596119885

minus 120579119872

) (minus2 120579119886119872119873

120572sin 120578119872

+ 119902119886119872119873

120572sin 120578119872

+ 120579119886119872119873

1205722cos 120578119872

) minus119886119872119873

cos 120578119872

120572

119898120572

119885120572119876119878ref119897ref

119869119885

(28)

1198912

= 120579119872

(( 120579119872

minus 119902) 119886119872119873

cos 120578119872

minus (120596119885

minus 120579119872

)119886119872119873

120572sin 120578119872

)

(29)

1198913

= 119902 (minus 120579119872

119886119872119873

cos 120578119872

+ 119902 (119886119872119873

cos 120578119872

minus 119886119879119873

cos 120578119879) minus 120579119879119886119879119873

cos 120578119879

+119886119872119873

120572(120596119885

minus 120579119872

) sin 120578119872

minus 119886119879sin 120578119879)

(30)

1198914

= 120579119879

(minus 119902119886119879cos 120578119879

minus 120579119879119886119879cos 120578119879

minus 119886119879sin 120578119879) (31)

1198915

= 119886119879

(minus 119902 sin 120578119879

minus 2 120579119879sin 120578119879) + 119886119879cos 120578119879 (32)

1198916

= 119902 (119886119872119873

sin 120578119872

minus 119886119879sin 120578119879

minus 2 119903) minus 4 119903 sdot119902 minus 3 119903 sdot 119902

minus 2119903 sdot 119902

(33)

120578119872

= 120579119872

minus 119902

120578119879

= 120579119879

+ 119902

(34)

In (26) it can be seen that the control input 120575119885appears

expressly in the third-order derivative of the control output119902 Therefore the relative degree of the control input 120575

119885is 3

3 The Quasi-Continuous High Order SlidingMode Controller

31 Sliding Mode Manifold Design To design the HOSMcontroller a sliding manifold must be chosen first In thisdesign we try to make the LOS rate converge to zero ora small neighbor domain near zero thus ensuring that themissile approaches its target in a quasi-parallel way whichwill lead to a minimal overload requirement So the slidingmanifold is chosen as follows

120590 = 119902 (35)

From the above discussion in Section 2 we know thatthe control input in relation to control output 119902 namelythe relative degree of sliding mode manifold 120590 is 3 So thefollowing design will be about a third-order sliding modecontroller

32 Design of the Quasi-Continuous HOSM Controller First(26) can be expressed as follows

120590 = ℎ (119905 119909) + 119892 (119905 119909) 119906 (36)

where ℎ(119905 119909) 119892(119905 119909) and 119906 are expressed as follows

ℎ (119905 119909) =1

119903(1198911

+ 1198912

+ 1198913

+ 1198914

+ 1198915

+ 1198916)

119892 (119905 119909) = 119891120575119885

119906 = 120575119885

(37)

According to the quasi-continuous high order slidingmode control method proposed by Levant in [15] the slidingmode manifold whose relative degree is 3 should be designedin the following form where 120573 is a control gain term

119906 = minus120573

+ 2 (|| + |120590|23

)minus12

( + |120590|23 sign120590)

|| + 2 (|| + |120590|23

)12

(38)

The conditions under which the LOS angular velocitymay converge are as follows

0 lt 119870119898

le 119892 (119905 119909) le 119870119872

|ℎ (119905 119909)| le 119862

(39)

where119870119898119870119872 and119862 are all larger than zeroThis is a proven

theorem by Levant in [15]The system we discussed meets the above requirements

and the proof is as followsEquation (27) shows the following

119892 (119905 119909) = 119891120575119885

= minus1

119903

119862120572

119884119876119878ref

119898

119898120575119885

119911119876119878ref119897ref

119869119885

cos (120579119872

minus 119902)

(40)

The dynamic pressure 119876 is 119876 = 1205881198812

1198722 where 120588 =

008803Kgm3 (altitude = 20Km) is the air density and119881119872

=

2000ms is the speed of the missile so 119876 is always positive119878ref = 026m2 and 119897ref = 365m denote the reference area

and the reference length of the missile they are both positiveconstant

119898 = 100Kg denotes the missile mass 119869119885

= 106m2 Kgdenotes the rotational inertia

119903 is the relative distance it is always a positive number119862120572

119884is the lift coefficient caused by the angle of attack it

varies from 018 to 037 and it is always a positive number119898120575119911

119911is the moment coefficient caused by the actuator

deflection In the normal layout (actuator lays behind thecenter of gravity) 119898

120575119911

119911is always negative

Meanwhile consider that the missile under guidance andcontrol is unlikely to fly away from its target namely the anglebetween themissilersquos velocity and its LOS direction cannot belarger than 90∘ then

1003816100381610038161003816120579119872 minus 1199021003816100381610038161003816 lt

120587

2

997904rArr cos (120579119872

minus 119902) gt 0

(41)


Summing up the above conditions then we can get

119892 (119905 119909) gt 0 (42)

In other words there is a positive real number 119870119898existing

that could satisfy the following condition

0 lt 119870119898

lt 119892 (119905 119909) (43)

Before the missile hits on the target the term will be positiveand limited then we can get

0 lt 119870119898

lt 119892 (119905 119909) lt 119870119872

(44)

With (37) then

ℎ (119905 119909) =1

119903(1198911

+ 1198912

+ 1198913

+ 1198914

+ 1198915

+ 1198916) (45)

In the practice sense the changes in both the LOS rateand the acceleration of the missile and the acceleration of thetarget are limited and continuous So the following variables119902 119902

119902 119886119872 119886119879 and 119886

119879are all bounded However because

ℎ(119905 119909) contains the item 1119903 when the relative distancebetween the missile and its target is zero the boundaryof ℎ(119905 119909) is not guaranteed In [15] Levant only requiresthat condition (39) should be locally valid not requiringthat it should be globally valid Therefore the integratedguidance and controlmethod is applicable here So the abovementioned condition is satisfied with a positive number 119862

|ℎ (119905 119909)| le 119862 (46)

33TheVirtual ControlDesign Whenusing the slidingmodecontrolmethod the avoidance of the chattering phenomenonhas always been a key issue being discussed In the tradi-tional method researchers in [16 17] have proposed severalsaturation functions to replace the sign functions to builda boundary layer to alleviate the chattering or to use fuzzylogic to displace the high-frequency switching term To ourknowledge none of these approaches has proven that therefined controller still retains their robustness against theuncertainties and disturbances In this work in order toalleviate the chattering phenomenon we do not directly usethe third-order controller but introduce the virtual control119906119894= 120575119885to perform the actual control

120575119885

= int 120575119885dt = int 119906

119894dt (47)

After the relative degree is increased to the fourth order weget the following expressions

120590(4)

= ℎlowast

(119905 119909) + 119892lowast

(119905 119909) 119906119894= ℎlowast

(119905 119909) + 119892 (119905 119909) 119906119894

ℎlowast

(119905 119909) = ℎ (119905 119909) + 119892 (119905 119909) 120575119885

119892 (119905 119909)

=119898120575119885

1199111198762

1198782

119897119862120572

119884

119869119885

1198981199032[( 119902 minus 120579

119872) 119903 sin 120578

119872+ 119903 cos 120578

119872]

(48)

Even though the expression of ℎ(119905 119909) is rather compli-cated it is still the function of 119902 119902 119902 119886

119872 119886119879 and 119886

119879 therefore

similar to ℎ(119905 119909) it has its boundary except themomentwhenthe missile hits on its target For the same reason 119892(119905 119909)

and the rudder deflection 120575119885also have their boundaries

Therefore we get the condition that |ℎlowast

(119905 119909)| le 119862 (119862 gt 0)Because 120575

119885is obtained through the derivation of 120575

119885 119892lowast(119905 119909)

is the same as 119892(119905 119909) thus 0 lt 119870119898

lt 119892lowast

(119905 119909) lt 119870119872

issatisfied

According to the formula of the fourth-order controllergiven by Levant in [15] we give the following formulae forthe virtual control 119906

119894

119906119894= minus120573

Φ34

11987334

Φ34

=120590 + 3 [||

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

minus12

sdot [

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

11987334

=1003816100381610038161003816

120590

1003816100381610038161003816 + 3 [||

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

minus12

sdot

1003816100381610038161003816100381610038161003816

+ (|| + 05 |120590|34

)minus13

( + 05 |120590|34 sign120590)

1003816100381610038161003816100381610038161003816

(49)

The integral of the virtual control 120575119885produces the actual

control command 120575119885 120590 in the controller is obtained with the

Arbitrary-Order Robust Exact Differentiator presented in thefollowing section

34 The Arbitrary-Order Robust Exact Differentiator Thequasi-continuous HOSM control method needs to use thethird derivative of the sliding manifold namely 119902

(4) Howto calculate or accurately estimate 119902

(4) is one of the keyproblems to be solved We use the Arbitrary-Order RobustExact Differentiator designed by Levant to differentiate theLOS rate 119902 thus obtaining 119902 119902 and 119902

(4)According to (44) and (45) the following condition is

valid1003816100381610038161003816

120590

1003816100381610038161003816 le 119862 + 120573119870119872

(50)

The Arbitrary-Order Robust Exact Differentiator can beconstructed in accordance with high order sliding modesdifferentiation and output feedback control in [18]

If a certain signal 119891(119905) is a function consisting of abounded Lebesgue-measurable noise with unknown base


signal 1198910(119905) whose 119903th derivative has a known Lipschitz

constant 119871 gt 0 then the 119899th-order differentiator is definedas follows

0

= V0

V0

= minus12058201198711(119899+1) 10038161003816100381610038161199110 minus 119891 (119905)

1003816100381610038161003816

119899(119899+1) sign (1199110

minus 119891 (119905))

+ 1199111

1

= V1

V1

= minus12058211198712(119899+1) 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

(119899minus1)119899 sign (1199111

minus V0) + 1199112

119899minus1

= V119899minus1

V119899minus1

= minus120582119899minus1

11987112 1003816100381610038161003816119911119899minus1 minus V

119899minus2

1003816100381610038161003816

12 sign (119911119899minus1

minus V119899minus2

)

+ 119911119899

119899

= minus120582119899119871 sign (119911

119899minus V119899minus1

)

(51)

and if 120582119894

gt 0 is sufficiently large the convergence is guaran-teed

To obtain the third-order derivative of 119902 we constructthe third-order sliding mode differentiator and estimate thederivative of 119902 for each order In view of differential precisionwe configure the following fifth-order differentiator SeeAppendix A for comparison

0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(52)

where 1199113 1199112 1199111 and 119911

0are the estimations of 119902

(4) 119902 119902 and 119902

4 Baseline Separated Guidance andControl Method

To verify the homing performance of the integrated guidanceand control method we compare it with the separatedguidance and control methodThe guidance laws we used arethe proportional navigation (PN) guidance law for attackinga nonmaneuvering target and the optimal sliding modeguidance law for attacking a maneuvering target

41 The Proportional Navigation (PN) Guidance Law Theguidance law has a simple formula and excellent perfor-mances for nonmaneuvering target Its formula is as follows

119899119888

= minus119873 119902119881119872

119892 (53)

119899119888is the overload command 119873 is the effective navigation

ratio 119902 is the LOS rate 119881119872

is the speed of the missile 119892 isthe acceleration of the gravity The guidance law gives theoverload command of the missile according to the LOS rateand then the controller gives the rudder deflection commandaccording to the overload command

42 The Optimal Sliding Mode Guidance Law The optimalsliding mode guidance law (OSMG) is a novel practicalguidance law proposed by D Zhou He combines the optimalguidance lawwith the slidingmode guidance law and designsthe new sliding mode guidance law that not only is robustto maneuvering target but also has the merits of the optimalguidance law such as good dynamic performance and energyconservation Its formula is as follows

119899119888

= minus3100381610038161003816100381610038160

10038161003816100381610038161003816119902 + 120576

119902

10038161003816100381610038161199021003816100381610038161003816 + 120575

(54)

where 119899119888is the overload command

0is the approach

velocity of the missile and its target 119902 is their LOS rate 120576 =

const is the compensatory gain 119902(| 119902| + 120575) is for substitutingfor sign( 119902) and for smoothing 120575 is a small quantity whichcould adjust the chattering

43 Separated Guidance and Control Design For simulationand comparisonwe use the conventional three-loop overloadautopilot as the controller which gives the rudder deflectioncommand according to the feedback of the three loops ofoverload pseudo-angle of attack and pitch rate The blockdiagram is as shown in Figure 3

As the figure shows the inner loop has the feedback onangular velocity which improves the damping characteristicsof the missile airframe

According to the aerodynamic coefficient of the missilewith selected working points we set 119870

119868= 019 119870

120572= 3 and

119870120596

= minus025 and the controller can well track the overloadcommand the rise time of its step response is 046 secondsand its settling time is 083 secondsThe step responses of themissile to overload command and the Bode diagram for openloop are shown in Figure 4


KIS

120596Z 120572 nY

nC K120596K120572 nY+minus+minus+minus dynamicsAirframe

modelServo

Figure 3 The working principles for three-loop overload autopilot

Step response

Time (s)0 02 04 06 08 1 12

0

02

04

06

08

1

Rise time (s) 0463

Settling time (s) 0831

Bode diagram

Frequency (rads)

To output pointFrom input pointTo output pointFrom input point

Gain margin (dB) 175 At frequency (rads) 15

Phase margin (deg) 709 At frequency (rads) 294

Am

plitu

de

Mag

nitu

de (d

B)Ph

ase (

deg)

0

minus180

minus360

minus54010410310210110010minus1

100

0

minus100

minus200

minus300

Figure 4 The autopilot performance step response and Bode diagram

5 Simulation Results

To verify the high order slidingmode integrated guidance andcontrol (HOSM-IGC) method we compare it with the base-line separated guidance and control Numerical simulationsare designed in three typical engagement scenarios

AORED parameters are as follows the initial value 1205820

=

1205821

= 1205822

= 1205823

= 50 1199110

= 01 1199111

= 1199112

= 1199113

= 0 119871 = 1400 thesimulation step is 00001 seconds

51 Scenario 1 Nonmaneuvering Target In the first scenariothe nonmaneuvering target does uniform rectilinear motionand PN guidance law with three-loop autopilot is introducedfor a comparison with the HOSM-IGCThe initial conditionsare set as shown in Table 1

The motion equations of the target are as follows

119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 0

(55)

The simulation results are shown in Figures 5 and 6The missilersquos flight trajectory and overload curve show

that within the first 3 sec the HOSM-IGC method spendsmuch energy (overload) on changing the initial LOSdirection

Table 1 Initial conditions

Variable Description Value(119883119872 119884119872) Missile initial position (0 km 18 km)

(119883119879 119884119879) Target initial position (60 km 20 km)

119881119872

Missile initial velocity 1200 (ms)119881119879

Target initial velocity 2000 (ms)120579119872

Missile initial flight path angle 0 (deg)120579119879

Target initial flight path angle 0 (deg)120573 Parameter of the controller 10 or 30119873 Navigation ratio of PN 3

of the missile After the boresight adjustment 120590 reaches thedesired sliding manifold and the missile does not need anymaneuver to hit the target This is because the target doesnot maneuver any more which means no disturbance isintroduced in the engagement dynamic so the state (or thesliding mode) of the missile will stay on the manifold Incontrast the overload command given by the PN guidancelaw increases fast as the relative distance decreases Toexamine the performances of the two guidance laws furtherwe increase the target speed and analyze the commandchanges

Therefore we set the target speeds 3000ms 4000msand carry out simulations The simulation result is shown inthe overload curve in Figure 7


175

180

185

190

195

200

205

TargetHOSM-IGCPN

Y(k

m)

0 10 20 30 40 50 60X (km)

Figure 5 Target and missile trajectories

0

5

10

15

20

25

0 5 10 15 20Time (s)

HOSM-IGCPN

Initial boresight adjustment

Sliding manifold reached

Miss

ile ac

cele

ratio

n (G

)

minus5

Figure 6 Missile acceleration profile

It is evident that the speed of divergence of the overloadcommand given by the PN guidance law increases withthe target speed Specifically when the target speed reaches4000ms the commanded overload is almost 30 g which isobviously not ideal for the attack of a nonmaneuvering targetbut with the HOSM-IGC method the missile adjusts itsboresight very quickly then maintains it around 0 g and fliesto its target in the rectilinear ballistic trajectory not affectedby the increases of target speed still accomplishing the high-precision hit-on collision

Figure 8 shows that although the HOSM-IGC methodachieves a more effective overload command on the otherside it sees some chattering when the sliding mode reaches

0

5

10

15

20

25

30

0 5 10 15Time (s)

Miss

ile ac

cele

ratio

n (G

)

minus5

HOSM-IGC (Vt = minus3000)HOSM-IGC (Vt = minus4000)

PN (Vt = minus3000)PN (Vt = minus4000)


0

1

2

0 5 10 15 20Time (s)

4 45 5

0

02

04

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 30)

Figure 8 Actuator deflection

the desired manifold the commanded rudder deflectionangle chatters for around 045 deg at about 15Hz which iskind of harmful to the system To reduce the chatteringwe adjust the controllerrsquos parameter 120573 = 10 and carry outsimulations again Figure 9 shows that after 120573 decreasesthe command of rudder deflection angle converges slower(for about 7 sec) however the chattering weakens obviouslyits magnitude being only about 015 degrees The smallerchattering well enhances convergence precision eventuallyreducing the target missing The reason that minor 120573 resultsin an alleviative chattering can be seen from (47) Figure 10shows the target and missile trajectories

Table 2 compares the average miss distance of 50 simula-tions under the conditions discussed above the comparison


Table 2 Average miss distance of 50 simulations

Target speed PN HOSM-IGC(120573 = 30)

HOSM-IGC(120573 = 10)

119881119905= 2000ms 115m 086 073m

119881119905= 3000ms 261m 117 106

119881119905= 4000ms 542m 156 134

0

1

2

0 5 10 15 20Time (s)

72 74 76 78

0

02

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)


results show that when the missile intercepts a nonmaneu-vering target the hit precision of the HOSM-IGC methodis apparently higher than that of the PN guidance law andthat the low-gain HOSM-IGC method can effectively reducethe chattering magnitude thus enhancing the interceptionprecision

52 Scenario 2 Step Maneuvering Target At the first stagethe target flies at uniform speed and in a rectilinear way after10 seconds it maneuvers at the normal acceleration of 5 g Inthis scenario OSMG is introduced for a comparison with theHOSM-IGC

119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

=

0 g 119905 lt 10 sec

5 g 119905 gt 10 sec

(56)

The initial simulation conditions are given in Table 3The overload curve in Figure 11 shows dearly that both

types of guidance laws can track the maneuvering targetDuring 0 to 10 seconds the target flies at uniform speed andin a rectilinear way and the missile converges its overload to




119881119872




Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001

15

17

19

21

23

25

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60X (km)


0 g and flies to its target after 10 seconds the target beginsto maneuver by 5 g Both types of guidance law respondby rapidly increasing the overload adjusting attitude andmaking the missilersquos boresight aim at its target

We can see that when both types of guidance law tracktheir targets the convergence speed of OSMG is almostthe same as the HOSM-IGC method But the HOSM-IGCmethod has higher convergence precision and needs loweroverload at the end phase

We can also see that after the target maneuvers if themissile is given enough time to track the targetrsquos maneuvernamely let the missilersquos overload command converge to theoverload of the target theremay not be largemiss distance Inother words for a certain period of time before the collisionthe targetmaneuver (it means only a limitedmaneuver whichdoes not include the condition that the maneuvering of thetarget for a long time may cause a change of the geometricalrelations between the missile and its target) has a small effecton both types of guidance law

But if themaneuver occurs rather late namely within oneto three seconds before collision when the overload com-mand of guidance law is not yet converged the approaching


0

5

10

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus5

minus10

Figure 11 Missile and target acceleration profile

0

10

20

30

0 5 10 15 20Time (s)

tgo = 1 stgo = 2 stgo = 3 s

tgo = 3 s (diverge)

tgo = 2 s (diverge)

tgo = 1 s (diverge)minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 10)


collision increases themiss distanceTherefore increasing theparameter 120573 can remarkably increase the convergence speedand effectively enhance guidance precision As Figure 12shows when 120573 = 10 the overload at the end phase convergesslowly even if the target maneuvers three seconds beforecollision the missilersquos overload still has no time to convergebeing unable to track themaneuvering target Figure 13 showsthat when 120573 increases to 30 and tgo = 3 seconds theoverload can converge to about 5 g but when tgo = 2 sec-onds the overload may continue to increase indicating that

0

10

20

30

0 5 10 15 20Time (s)


tgo = 1 s (diverge)

tgo = 2 s (diverge)

tgo = 3 s (converge)

minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

120573 = 30)HOSM-IGC (


0

10

20

30

0 5 10 15 20Time (s)

tgo = 1 s (diverge)




minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 50)


the convergence still needs to be quickened thus continuingto rise 120573 to 50 Figure 14 shows that the HOSM-IGC methodcan track the target that maneuvers when tgo = 2 secondsbut it causes the divergence of overload and the increase ofmiss distance if the target maneuvers when tgo = 1 secondFigure 15 gives the overload curve of the OSMG guidance lawand shows that the maneuver of the target before collisionmay cause the large-scale oscillation of the missilersquos overloadwhich may diverge to a large numerical value when thecollision occurs in the end


Table 4 Average miss distances of 50 simulations

Targetmaneuveringtiming

HOSM-IGC120573 = 30

HOSM-IGC120573 = 40

HOSM-IGC120573 = 50

OSMG

tgo = 1second 43539 3224 28936 36116

tgo = 2seconds 25424 18665 09322 35534

tgo = 3seconds 08124 08265 07538 11959

Time (s)0 5 10 15 20

0

10

20

30OSMG


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)


The analysis in Figures 12 13 14 and 15 shows that theoverload of the missile converges faster and its miss distanceis smaller with increasing 120573 To verify this finding we carryout 50 times Monte Carlo simulations in which the positionand speed of the target have 1 of random difference

The average miss distances are shown in Table 4 andFigure 16 Clearly the timing of the targetrsquos step maneuverdramatically affects the final interception precision morespecifically given a shorter reaction time for the guidanceand control system the missile seems more likely to missthe target To the OSMG guidance law in all three scenarioshardly does it show any advantages against theHOSM-IGC Itcan also be seen that with the increasing of the120573 theHOSM-IGC system responds even faster which leads to an obviousdecrease of the average miss distance The effect of 120573 on theresponse of the HOSM-IGC system is a valuable guidelinewhen implementing the proposed method into practice

53 Scenario 3 Weaving Target In this scenario the targetmaneuvers by 119886

119879= 40sin(1205871199052) OSMG with three-loop




119881119872





autopilot is introduced for comparison The motion equa-tions of the target are as follows

119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 40 sin(120587119905

2)

(57)

The initial simulation conditions are given in Table 5The missilersquos trajectory under the two guidance and

control methods given in Figure 17 does not see muchdifference

However the overload curve given in Figure 18 showsthat after the missile completes its initial attitude adjustmentwith the HOSM-IGC method it can almost ideally track itsmaneuvering target by contrast with the OSMG methodthe missile seems to have the tendency to track its targetrsquosmaneuver but has larger tracking errors Besides with theOSMGmethod the missilersquos overload increases rapidly at theend of attack primarily because of the divergence of its LOSrate On the other hand with the HOSM-IGC method themissile has no divergence even at the end of attack ensuringa smaller target missing quantity

The actuator deflection curve in Figure 19 shows thatin order to provide a rather big normal overload for theend phase the OSMG method produces a rather big rudderdeflection command however it may increase the missilersquostarget missing quantity once its rudder deflection saturatesand themissile does not have enough overloads or the ruddercannot respond that fast

As shown in Figure 20 because of the dramatic changein overload command the response of the missilersquos autopilotto high-frequency command sees an obvious phase lag andamplitude value attenuation its actual overload cannot trackthe command ideally this is a main reason why the missdistance increases However the controller in the HOSM-IGC method gives its rudder deflection command directlyand there is no lagging or attenuation caused by the autopilotthus enhancing the guidance precision effectively Further-more the fast convergence of the high order sliding modemakes the missile rapidly track its maneuvering target withthemost reasonable rudder deflection command reducing itsoverload effectively


0

05

1

15

2

25

3

35

4

45

5

0 10 20 30 40 50Simulation times

OSMG

Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


HOSM-IGC (120573 = 40)

HOSM-IGC (120573 = 50)

HOSM-IGC (120573 = 30)

Figure 16 Miss distances of HOSM-IGC (120573 = 30 40 50) and OSMG with target maneuvers at different time-to-go (tgo = 1 second 2seconds and 3 seconds)

The average miss distance of 50 simulations under theconditions is 073m for HOSM-IGC and 186m for OSMGWe can see that the HOSM-IGC method not only doesprovide a more reasonable actuator deflection command butalso achieves a higher interception precision

6 Conclusions

This paper proposes an LOS feedback integrated guidanceand control method using quasi-continuous high order

sliding mode guidance and control method With the fastand precise convergence of the quasi-continuous HOSMmethod the HOSM-IGCmethod performsmuch better thanthe traditional separated guidance and control method withless acceleration effort and less miss distance in all thethree simulation scenarios of nonmaneuvering target stepmaneuvering target and weaving target In addition the ideaof virtual control largely alleviates the chattering withoutany sacrifice of robustness As a result of the alleviationof the chattering the control input command 120575

119885becomes


175

180

185

190

195

200

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60 0 20 40X (km)

Figure 17 The trajectories of the missile and its target

0

10

20

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus10

minus20


smooth enough for the actuator to be implemented as aresult the interception accuracy is improved In designingthe quasi-continuous sliding mode controller the Arbitrary-Order Robust Exact Differentiator is introduced to estimatethe high order time derivative of the LOS rate 119902 which notonly ensures the guidance and control precision but alsosimplifies the calculation

0

10

20

30

0 5 10 15 20Time (s)

OSMGHOSM-IGC

minus10

minus20

minus30

minus40

Actu

ator

defl

ectio

n (d

eg)


0

50

100

150

200

15 16 17 18Time (s)

Commanded accelerationAchieved acceleration

Miss

ile ac

cele

ratio

n (G

)

minus50

Figure 20 Commanded acceleration and achieved acceleration

Appendices

A The Third-Order RobustExact Differentiator

The third-order robust exact differentiator built according tothe following parameters and equations is used to performthe simulation the initial value 120582

0= 1205821

= 1205822

= 1205823

= 50


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0

100

200

300

400

0 02 04Time (s)

h(t)

z3

h(t) and z3

minus100

minus200

minus300

minus400

minus5000 02 04

Time (s)

g(t)

z2

g(t) and z2

minus100

minus200

minus300

minus400

f(t)

z1

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

F(t)

z0

F(t) and z0

minus0005

minus001

Figure 21 The outputs of the differentiator in the sampling interval of 0001 seconds

the initial values 1199110

= 01 1199111

= 1199112

= 1199113

= 0 119871 = 1400 thesimulation step is 0001 seconds Consider

0

= V0

V0

= minus120582011987114 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

34 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987113 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

23 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987112 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

12 sign (1199112

minus V1) + 1199113

3

= minus1205823119871 sign (119911

3minus V2)

(A1)


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0 02 04Time (s)

0

200

400

0 02 04Time (s)

h(t) and z3

minus200

minus400

minus600

minus800

g(t) and z2

minus100

minus200

minus300

minus400

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

minus16

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0


If the designed signals are 119865(119905) = int 119891(119905)119889119905 119891(119905) =

int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +

3cos(120587119905) then theoretically 119891(119905) is the first-order differ-ential of input signal 119892(119905) is the second-order differ-ential of input signal ℎ(119905) is the third-order differen-tial of input signal Through the simulation the differ-ential signal of actual output is compared with that of

theoretical output the comparison results are shown inFigure 21

Figure 21 shows that 1199110can effectively track the signal

119865(119905) but 1199112has obvious errors in tracking the second-order

differential 119892(119905) 1199113has greater errors in tracking the third-

order differential thus the overall differential effect is notideal


0 02 04

0

0005

001

0015

002

0025

003

0035

004

Time (s)0 02 04

0

01

02

03

04

05

Time (s)

0 02 04

0

05

1

15

2

Time (s)0 02 04

0

05

1

15

2

25

3

35

4

Time (s)

h(t) and z3g(t) and z2

minus04

minus05

f(t) and z1

minus2

minus15

minus1

minus05

minus1

minus05

minus01

minus02

minus03

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0


There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere

Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows

10038161003816100381610038161003816120590(119894)

10038161003816100381610038161003816le 120583120591119903minus119894

119894 = 0 119903 minus 1 (A2)

That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision

Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22


Table 6 Tracking errors of the AORED with different orders andsampling interval

Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001

First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005

As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911

2has an obviously

smaller tracking error in the second-order differential 119892(119905)

tracking in the beginning 1199113has a rather sharp peak in the

third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator

B The Fifth-Order Robust Exact Differentiator

The settings of the fifth-order differentiator are given asfollows

0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)

The initial value 1199110

= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825

= 50 the initial values 1199111 1199112 1199113 1199114 1199115

= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23

As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)

To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting

Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2

Diameter 0178mMass 1016 Kg119868119885

1063 Kgsdotm2

an appropriate sampling interval and using the differentiatorwith a relatively high number of orders

C Physical and Geometric Characteristics

See Table 7

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of

a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999

[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012

[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014

[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014

[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867

infincontrol frameworksrdquo

IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003

[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design

for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001

[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003

[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007

[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE


Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006

[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006

[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009

[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013

[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008

[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005

[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014

[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014

[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Being famous for its good robustness to bounded distur-bances the sliding mode control method is introduced intothe study of guidance law In [2] Zhou et al gave the con-ditions for the sliding mode motion of a linear time-varyingsystem not to be disturbed by the disturbances and param-eter perturbations and proposed an adaptive sliding modeguidance law (ASMG) simulation results demonstrated thatthe ASMG is robust to uncertainties like target accelerationTo get finite time convergence modified SMC like terminalsliding mode control (TSMC) is proposed In [3] Zeng andHu combine the advantages of linear and terminal slidingmode controls which guarantee the convergence of trackingerrors in finite time And then nonsingular terminal slidingmode control (NTSMC) is introduced in the guidance lawdesign In [4] Kumar et al proposed a nonsingular terminalsliding mode guidance law with finite time convergencewhich avoids the singularity that may lead to the saturationof the control Integral SMC (ISMC) like that introduced in[5] is another revised sliding mode control which introducedthe integral sliding mode scheme to the guidance law designthe proposed ISMC guidance law provided a smaller controlmagnitude than the traditional sliding mode design Theadvance of the 119867

infincontroller greatly enriched the methods

to implement a robust control system In [6] Savkin etal suitably modified the 119867

infincontrol theory and provided

an effective framework for the precision missile guidanceproblem which showed much better performance than thelinear quadratic optimal guidance law in the computersimulation In [7] Chen and Yang proposed a mixed 119867

2119867infin

guidance design against maneuvering targets in which thecomplete nonlinear kinematics of the pursuit-evasionmotionwas considered

In addition to the separated guidance and control lawthe integrated guidance and control method is under heateddiscussion According to the relative motion between thetarget and the missile the traditional guidance law calculatesthe overload needed to hit the target and inputs the overloadcommand into the overload autopilot while the integratedguidance and control (IGC) method gives the rudder deflec-tion command directly to the missile according to therelative motion which evidently responds more quickly In[8] Menon et al designed an IGCmethod by using the linearquadratic optimal theory but the robustness of the system isnot satisfactory In [9] Vaddi et al provided a fully numericalmethodology for deriving state-dependent Riccati equationcontrollers for arbitrarily complex dynamic systems andapplied it in the IGC design of a missile Simulation resultsdemonstrated the effectiveness of the method In [10] Xinet al employed the theta-D method to give an approximateclosed-form suboptimal feedback controller to the nonlinearinfinite-horizon IGC problem Taking another look slidingmode approaches are also employed in the IGC design ofhoming missiles In [11] Shima et al proposed the slidingmode integrated guidance and control method based on theZEM Shtessel and Tournes doesmassive research on the highorder sliding mode controller design and proposed his newmethod In [12] he designed the high order sliding modeguidance law based on the smooth second-order slidingmode control (SSOSMC) which is smoother in high orders

than the traditional second-order sliding mode guidance lawIn [13] based on the geometrical homogeneity theory Donget al designed the tranquility control law for the integratedguidance and control model which however has rathermore parameters and is sensitive to parameters and theparameters must be carefully selected In [14] Mingzhe andGuangren used the sliding mode control theory to design theadaptive nonlinear feedback controller which is primarily forfixed ground target being unable to deal with the vehementperturbation caused by target maneuvering

Motivated by the aforementioned considerations thiswork will design an IGC scheme for the interception of thenear space maneuvering hypersonic vehicles Firstly a line-of-sight (LOS) rate feedback scheme is adopted to derive theIGC law and as the relative order of the control input to theLOS rate is higher than one a HOSM approach is introducedSecondly to implement the HOSM approach the 119899 timesderivations of the sliding manifold 119878 119878

119878 119878

(119899) must beknown However reconstruction of each 119878 119878

119878 119878

(119899) bytheir analytical expression could be rather difficult in prac-tice An alternative way adopted in this paper is to use theArbitrary-Order Robust Exact Differentiator (AORED) toestimate the derivations And next to alleviate the chatteringphenomenon caused by the HOSM controller the idea ofvirtual control is introducedThe virtual control V is designedand used as the control input of a system extended from theoriginal one and the real control 119906 acting on the real systemis obtained by integrating the virtual control V Benefittingfrom the integration element the real control input 119906 couldbe smooth enough for the implementation without reducingthe robustness of the HOSM method Finally the proposedmethod is implemented in a 3-dof model

The remaining part of this paper is organized as followsIn Section 2 the IGC model is derived in the longitudinalplane In Section 3 the quasi-continuous HOSM controlleras well as the AORED is designed In Section 4 the baselineseparated guidance laws and controller are given In Sec-tion 5 the numerical simulations are demonstrated in threetypical engagement scenarios And conclusions are made inSection 6

2 Integrated Guidance and Control Model

The traditional guidance and control algorithm usually usesthe guidance loop as its outer loop and is only responsible forgiving commanded overload then the control loop is onlyresponsible for tracking the overload command eventuallyachieving the missilersquos guidance toward its target Althoughit is always desirable to design the control loop or theautopilot to have better dynamic performance in actualitythe controller always has some delay and attenuation Asa result the missile always has some error in acting theoverload command That is one of the reasons why aninterceptor misses its target

The integrated guidance and control algorithm combinesthe guidance and control loops into one loop and avoidsthe delay and attenuation caused by them Its architecture isshown in Figure 1


Seeker

Guidancefiltering


ActuatorMissile

airframe

Missile position

Target position


Target acceleration

Guidancelaw



120575CnC

q

120575120575C

q

q nm


O

Y1

X1

M

aM

xb

120572120579M

q

VM

T

VT

120579T

aT

120599





119889119881119872

119889119905=

1


119872) (1)

119889120579119872

119889119905=

1

119898119881119872

(119875 sin120572 + 119884 minus 119898119892 cos 120579119872

) (2)

119889119909119872

119889119905= 119881119872cos 120579119872

(3)

119889119910119872

119889119905= 119881119872sin 120579119872

(4)

119889120596119885

119889119905=

119872119885

119869119885

(5)

119889120599

119889119905= 120596119885 (6)

120572 = 120599 minus 120579 (7)

where (119909119872

119910119872


119872 and its






119889119909119879

119889119905= minus119881119879cos 120579119879

119889119910119879

119889119905= 119881119879sin 120579119879

119889120579119879

119889119905=

119886119879119873

119881119879

(8)


119889119903

119889119905= minus119881119872cos (120579

119872minus 119902) minus 119881

119879cos (120579

119879+ 119902) (9)

119903119889119902

119889119905= minus119881119872sin (120579119872

minus 119902) + 119881119879sin (120579119879

+ 119902) (10)






119889119881119872

119889119905= 0

119889120579119872

119889119905=

119884

119898119881119872

(11)

Denote that 119886119872119873


119886119879119873


120579119872

=119886119872119873

119881119872

120579119879

=119886119879119873

119881119879

(12)



119884 = 119862120572

119884120572119876119878ref + 119862

120575119885

119884120575119885119876119878ref (13)

119872119885

= 119898120575119885

119885120575119885119876119878ref119897 + 119898

120572

119885120572119876119878ref119897 (14)

where119862120572


of attack 119862120575119885




119885is orders of


119884 = 119862120572

119884120572119876119878ref (15)

Then

119886119872119873

=119884

119898=

119862120572

119884120572119876119878ref

119898 (16)


119889120579119872

119889119905=

119862120572

119884120572119876119878ref

119898119881119872

(17)

119889120596119885

119889119905=

119898120575119885

119885120575119885

119876119878ref119897 + 119898120572

119885120572119876119878ref119897

119869119885

(18)

119889120599

119889119905= 120596119885 (19)

120572 = 120599 minus 120579119872

(20)


( 119902 120579119872



119902 = minus2119903

119903sdot 119902 minus

119862120572

119884119876119878ref

119898119903cos (120579

119872minus 119902) sdot 120572

+119886119879119873

119903cos (120579

119879+ 119902)

119903 119902 = minus119881119872sin (120579119872

minus 119902) + 119881119879sin (120579119879

+ 119902)

120579119872

=1

119898119881119872

119862120572

119884119876119878ref sdot 120572

= 120596119885

minus1

119898119881119872

119862120572

119884119876119878ref sdot 120572

119885

=119898120575119885

119885119876119878ref119897

119869119885

sdot 120575119885

+119898120572

119885119876119878ref119897

119869119885

120572

(21)



=1

119903[minus2 119903 119902 minus 119886

119872119873cos (120579

119872minus 119902) + 119886

119879119873cos (120579

119879+ 119902)]

(22)



119902 =

1


119872119873( 119902 minus 120579

119872) sin (120579

119872minus 119902)

+ 119886119872119873

cos (120579119872

minus 119902)] minus [119886119879119873

( 119902 + 120579119879) sin (120579

119879+ 119902)

minus 119886119879119873

cos (120579119879

+ 119902)]

(23)



119886119872119873

=119862120572

119884119876119878ref

119898 =

119862120572

119884119876119878ref

119898( 120599 minus 120579

119872)

=119862120572

119884119876119878ref

119898(120596119885

minus 120579119872

)

(24)




119872as follows

119886119872119873

=119862120572

119884119876119878ref

119898(119885

minus119886119872

119881119872

)

=119862120572

119884119876119878ref

119898

119898120575119885

119885119876119878ref119897ref

119869119885

sdot 120575119885

+119862120572

119884119876119878ref

119898(

119898120572

119885119876119878ref119897ref

119869119885

120572 minus119886119872

119881119872

)

(25)


Thus

119902(4)

= 119891120575119885

sdot 120575119885

+1

119903(1198911

+ 1198912

+ 1198913

+ 1198914

+ 1198915

+ 1198916) (26)

where

119891120575119885

= minus1

119903

119862120572

119884119876119878ref cos 120578

119872

119898

119898120575119885

119885119876119878ref119897ref

119869119885

(27)

1198911

= (120596119885

minus 120579119872

) (minus2 120579119886119872119873

120572sin 120578119872

+ 119902119886119872119873

120572sin 120578119872

+ 120579119886119872119873

1205722cos 120578119872

) minus119886119872119873

cos 120578119872

120572

119898120572

119885120572119876119878ref119897ref

119869119885

(28)

1198912

= 120579119872

(( 120579119872

minus 119902) 119886119872119873

cos 120578119872

minus (120596119885

minus 120579119872

)119886119872119873

120572sin 120578119872

)

(29)

1198913

= 119902 (minus 120579119872

119886119872119873

cos 120578119872

+ 119902 (119886119872119873

cos 120578119872

minus 119886119879119873

cos 120578119879) minus 120579119879119886119879119873

cos 120578119879

+119886119872119873

120572(120596119885

minus 120579119872

) sin 120578119872

minus 119886119879sin 120578119879)

(30)

1198914

= 120579119879

(minus 119902119886119879cos 120578119879

minus 120579119879119886119879cos 120578119879

minus 119886119879sin 120578119879) (31)

1198915

= 119886119879

(minus 119902 sin 120578119879

minus 2 120579119879sin 120578119879) + 119886119879cos 120578119879 (32)

1198916

= 119902 (119886119872119873

sin 120578119872

minus 119886119879sin 120578119879


minus 2119903 sdot 119902

(33)

120578119872

= 120579119872

minus 119902

120578119879

= 120579119879

+ 119902

(34)



119885is 3



120590 = 119902 (35)



120590 = ℎ (119905 119909) + 119892 (119905 119909) 119906 (36)


ℎ (119905 119909) =1

119903(1198911

+ 1198912

+ 1198913

+ 1198914

+ 1198915

+ 1198916)

119892 (119905 119909) = 119891120575119885

119906 = 120575119885

(37)


119906 = minus120573

+ 2 (|| + |120590|23

)minus12

( + |120590|23 sign120590)

|| + 2 (|| + |120590|23

)12

(38)


0 lt 119870119898

le 119892 (119905 119909) le 119870119872

|ℎ (119905 119909)| le 119862

(39)




119892 (119905 119909) = 119891120575119885

= minus1

119903

119862120572

119884119876119878ref

119898

119898120575119885

119911119876119878ref119897ref

119869119885

cos (120579119872

minus 119902)

(40)


1198722 where 120588 =


=










120575119911



1003816100381610038161003816120579119872 minus 1199021003816100381610038161003816 lt

120587

2

997904rArr cos (120579119872

minus 119902) gt 0

(41)



119892 (119905 119909) gt 0 (42)



0 lt 119870119898

lt 119892 (119905 119909) (43)


0 lt 119870119898

lt 119892 (119905 119909) lt 119870119872

(44)

With (37) then

ℎ (119905 119909) =1

119903(1198911

+ 1198912

+ 1198913

+ 1198914

+ 1198915

+ 1198916) (45)


119902 119886119872 119886119879 and 119886



|ℎ (119905 119909)| le 119862 (46)


120575119885

= int 120575119885dt = int 119906

119894dt (47)


120590(4)

= ℎlowast

(119905 119909) + 119892lowast

(119905 119909) 119906119894= ℎlowast

(119905 119909) + 119892 (119905 119909) 119906119894

ℎlowast

(119905 119909) = ℎ (119905 119909) + 119892 (119905 119909) 120575119885

119892 (119905 119909)

=119898120575119885

1199111198762

1198782

119897119862120572

119884

119869119885

1198981199032[( 119902 minus 120579

119872) 119903 sin 120578

119872+ 119903 cos 120578

119872]

(48)


119872 119886119879 and 119886

119879 therefore




(119905 119909)| le 119862 (119862 gt 0)Because 120575


119885 119892lowast(119905 119909)


lt 119892lowast

(119905 119909) lt 119870119872

issatisfied


119894

119906119894= minus120573

Φ34

11987334

Φ34

=120590 + 3 [||

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

minus12

sdot [

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

11987334

=1003816100381610038161003816

120590

1003816100381610038161003816 + 3 [||

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

minus12

sdot

1003816100381610038161003816100381610038161003816

+ (|| + 05 |120590|34

)minus13

( + 05 |120590|34 sign120590)

1003816100381610038161003816100381610038161003816

(49)








valid1003816100381610038161003816

120590

1003816100381610038161003816 le 119862 + 120573119870119872

(50)






0

= V0

V0

= minus12058201198711(119899+1) 10038161003816100381610038161199110 minus 119891 (119905)

1003816100381610038161003816

119899(119899+1) sign (1199110

minus 119891 (119905))

+ 1199111

1

= V1

V1

= minus12058211198712(119899+1) 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

(119899minus1)119899 sign (1199111

minus V0) + 1199112

119899minus1

= V119899minus1

V119899minus1


11987112 1003816100381610038161003816119911119899minus1 minus V

119899minus2

1003816100381610038161003816

12 sign (119911119899minus1

minus V119899minus2

)

+ 119911119899

119899

= minus120582119899119871 sign (119911


)

(51)

and if 120582119894



0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(52)

where 1199113 1199112 1199111 and 119911


(4) 119902 119902 and 119902




119899119888

= minus119873 119902119881119872

119892 (53)





119899119888

= minus3100381610038161003816100381610038160

10038161003816100381610038161003816119902 + 120576

119902

10038161003816100381610038161199021003816100381610038161003816 + 120575

(54)


0is the approach






119868= 019 119870

120572= 3 and

119870120596



KIS

120596Z 120572 nY


modelServo


Step response

Time (s)0 02 04 06 08 1 12

0

02

04

06

08

1

Rise time (s) 0463


Bode diagram

Frequency (rads)




Am

plitu

de

Mag

nitu

de (d

B)Ph

ase (

deg)

0

minus180

minus360

minus54010410310210110010minus1

100

0

minus100

minus200

minus300





=

1205821

= 1205822

= 1205823

= 50 1199110

= 01 1199111

= 1199112

= 1199113




119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 0

(55)






119881119872








175

180

185

190

195

200

205

TargetHOSM-IGCPN

Y(k

m)

0 10 20 30 40 50 60X (km)


0

5

10

15

20

25

0 5 10 15 20Time (s)

HOSM-IGCPN



Miss

ile ac

cele

ratio

n (G

)

minus5




0

5

10

15

20

25

30

0 5 10 15Time (s)

Miss

ile ac

cele

ratio

n (G

)

minus5




0

1

2

0 5 10 15 20Time (s)

4 45 5

0

02

04

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 30)







HOSM-IGC(120573 = 10)

119881119905= 2000ms 115m 086 073m

119881119905= 3000ms 261m 117 106

119881119905= 4000ms 542m 156 134

0

1

2

0 5 10 15 20Time (s)

72 74 76 78

0

02

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)




119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

=

0 g 119905 lt 10 sec

5 g 119905 gt 10 sec

(56)






119881119872





15

17

19

21

23

25

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60X (km)







0

5

10

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus5

minus10


0

10

20

30

0 5 10 15 20Time (s)


tgo = 3 s (diverge)

tgo = 2 s (diverge)


minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 10)



0

10

20

30

0 5 10 15 20Time (s)


tgo = 1 s (diverge)

tgo = 2 s (diverge)


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

120573 = 30)HOSM-IGC (


0

10

20

30

0 5 10 15 20Time (s)

tgo = 1 s (diverge)




minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 50)






HOSM-IGC120573 = 30

HOSM-IGC120573 = 40

HOSM-IGC120573 = 50

OSMG

tgo = 1second 43539 3224 28936 36116

tgo = 2seconds 25424 18665 09322 35534

tgo = 3seconds 08124 08265 07538 11959

Time (s)0 5 10 15 20

0

10

20

30OSMG


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)









119881119872






119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 40 sin(120587119905

2)

(57)







0

05

1

15

2

25

3

35

4

45

5


OSMG

Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


HOSM-IGC (120573 = 40)

HOSM-IGC (120573 = 50)

HOSM-IGC (120573 = 30)



6 Conclusions



119885becomes


175

180

185

190

195

200

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60 0 20 40X (km)


0

10

20

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus10

minus20



0

10

20

30

0 5 10 15 20Time (s)

OSMGHOSM-IGC

minus10

minus20

minus30

minus40

Actu

ator

defl

ectio

n (d

eg)


0

50

100

150

200

15 16 17 18Time (s)


Miss

ile ac

cele

ratio

n (G

)

minus50


Appendices



0= 1205821

= 1205822

= 1205823

= 50


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0

100

200

300

400

0 02 04Time (s)

h(t)

z3

h(t) and z3

minus100

minus200

minus300

minus400

minus5000 02 04

Time (s)

g(t)

z2

g(t) and z2

minus100

minus200

minus300

minus400

f(t)

z1

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

F(t)

z0

F(t) and z0

minus0005

minus001



= 01 1199111

= 1199112

= 1199113


0

= V0

V0

= minus120582011987114 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

34 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987113 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

23 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987112 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

12 sign (1199112

minus V1) + 1199113

3

= minus1205823119871 sign (119911

3minus V2)

(A1)


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0 02 04Time (s)

0

200

400

0 02 04Time (s)

h(t) and z3

minus200

minus400

minus600

minus800

g(t) and z2

minus100

minus200

minus300

minus400

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

minus16

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0



int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +








0 02 04

0

0005

001

0015

002

0025

003

0035

004

Time (s)0 02 04

0

01

02

03

04

05

Time (s)

0 02 04

0

05

1

15

2

Time (s)0 02 04

0

05

1

15

2

25

3

35

4

Time (s)


minus04

minus05

f(t) and z1

minus2

minus15

minus1

minus05

minus1

minus05

minus01

minus02

minus03

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0




10038161003816100381610038161003816120590(119894)

10038161003816100381610038161003816le 120583120591119903minus119894

119894 = 0 119903 minus 1 (A2)








2has an obviously






0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)


= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825





Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2


1063 Kgsdotm2



See Table 7



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Seeker

Guidancefiltering


ActuatorMissile

airframe

Missile position

Target position


Target acceleration

Guidancelaw



120575CnC

q

120575120575C

q

q nm


O

Y1

X1

M

aM

xb

120572120579M

q

VM

T

VT

120579T

aT

120599





119889119881119872

119889119905=

1


119872) (1)

119889120579119872

119889119905=

1

119898119881119872

(119875 sin120572 + 119884 minus 119898119892 cos 120579119872

) (2)

119889119909119872

119889119905= 119881119872cos 120579119872

(3)

119889119910119872

119889119905= 119881119872sin 120579119872

(4)

119889120596119885

119889119905=

119872119885

119869119885

(5)

119889120599

119889119905= 120596119885 (6)

120572 = 120599 minus 120579 (7)

where (119909119872

119910119872


119872 and its






119889119909119879

119889119905= minus119881119879cos 120579119879

119889119910119879

119889119905= 119881119879sin 120579119879

119889120579119879

119889119905=

119886119879119873

119881119879

(8)


119889119903

119889119905= minus119881119872cos (120579

119872minus 119902) minus 119881

119879cos (120579

119879+ 119902) (9)

119903119889119902

119889119905= minus119881119872sin (120579119872

minus 119902) + 119881119879sin (120579119879

+ 119902) (10)






119889119881119872

119889119905= 0

119889120579119872

119889119905=

119884

119898119881119872

(11)

Denote that 119886119872119873


119886119879119873


120579119872

=119886119872119873

119881119872

120579119879

=119886119879119873

119881119879

(12)



119884 = 119862120572

119884120572119876119878ref + 119862

120575119885

119884120575119885119876119878ref (13)

119872119885

= 119898120575119885

119885120575119885119876119878ref119897 + 119898

120572

119885120572119876119878ref119897 (14)

where119862120572


of attack 119862120575119885




119885is orders of


119884 = 119862120572

119884120572119876119878ref (15)

Then

119886119872119873

=119884

119898=

119862120572

119884120572119876119878ref

119898 (16)


119889120579119872

119889119905=

119862120572

119884120572119876119878ref

119898119881119872

(17)

119889120596119885

119889119905=

119898120575119885

119885120575119885

119876119878ref119897 + 119898120572

119885120572119876119878ref119897

119869119885

(18)

119889120599

119889119905= 120596119885 (19)

120572 = 120599 minus 120579119872

(20)


( 119902 120579119872



119902 = minus2119903

119903sdot 119902 minus

119862120572

119884119876119878ref

119898119903cos (120579

119872minus 119902) sdot 120572

+119886119879119873

119903cos (120579

119879+ 119902)

119903 119902 = minus119881119872sin (120579119872

minus 119902) + 119881119879sin (120579119879

+ 119902)

120579119872

=1

119898119881119872

119862120572

119884119876119878ref sdot 120572

= 120596119885

minus1

119898119881119872

119862120572

119884119876119878ref sdot 120572

119885

=119898120575119885

119885119876119878ref119897

119869119885

sdot 120575119885

+119898120572

119885119876119878ref119897

119869119885

120572

(21)



=1

119903[minus2 119903 119902 minus 119886

119872119873cos (120579

119872minus 119902) + 119886

119879119873cos (120579

119879+ 119902)]

(22)



119902 =

1


119872119873( 119902 minus 120579

119872) sin (120579

119872minus 119902)

+ 119886119872119873

cos (120579119872

minus 119902)] minus [119886119879119873

( 119902 + 120579119879) sin (120579

119879+ 119902)

minus 119886119879119873

cos (120579119879

+ 119902)]

(23)



119886119872119873

=119862120572

119884119876119878ref

119898 =

119862120572

119884119876119878ref

119898( 120599 minus 120579

119872)

=119862120572

119884119876119878ref

119898(120596119885

minus 120579119872

)

(24)




119872as follows

119886119872119873

=119862120572

119884119876119878ref

119898(119885

minus119886119872

119881119872

)

=119862120572

119884119876119878ref

119898

119898120575119885

119885119876119878ref119897ref

119869119885

sdot 120575119885

+119862120572

119884119876119878ref

119898(

119898120572

119885119876119878ref119897ref

119869119885

120572 minus119886119872

119881119872

)

(25)


Thus

119902(4)

= 119891120575119885

sdot 120575119885

+1

119903(1198911

+ 1198912

+ 1198913

+ 1198914

+ 1198915

+ 1198916) (26)

where

119891120575119885

= minus1

119903

119862120572

119884119876119878ref cos 120578

119872

119898

119898120575119885

119885119876119878ref119897ref

119869119885

(27)

1198911

= (120596119885

minus 120579119872

) (minus2 120579119886119872119873

120572sin 120578119872

+ 119902119886119872119873

120572sin 120578119872

+ 120579119886119872119873

1205722cos 120578119872

) minus119886119872119873

cos 120578119872

120572

119898120572

119885120572119876119878ref119897ref

119869119885

(28)

1198912

= 120579119872

(( 120579119872

minus 119902) 119886119872119873

cos 120578119872

minus (120596119885

minus 120579119872

)119886119872119873

120572sin 120578119872

)

(29)

1198913

= 119902 (minus 120579119872

119886119872119873

cos 120578119872

+ 119902 (119886119872119873

cos 120578119872

minus 119886119879119873

cos 120578119879) minus 120579119879119886119879119873

cos 120578119879

+119886119872119873

120572(120596119885

minus 120579119872

) sin 120578119872

minus 119886119879sin 120578119879)

(30)

1198914

= 120579119879

(minus 119902119886119879cos 120578119879

minus 120579119879119886119879cos 120578119879

minus 119886119879sin 120578119879) (31)

1198915

= 119886119879

(minus 119902 sin 120578119879

minus 2 120579119879sin 120578119879) + 119886119879cos 120578119879 (32)

1198916

= 119902 (119886119872119873

sin 120578119872

minus 119886119879sin 120578119879


minus 2119903 sdot 119902

(33)

120578119872

= 120579119872

minus 119902

120578119879

= 120579119879

+ 119902

(34)



119885is 3



120590 = 119902 (35)



120590 = ℎ (119905 119909) + 119892 (119905 119909) 119906 (36)


ℎ (119905 119909) =1

119903(1198911

+ 1198912

+ 1198913

+ 1198914

+ 1198915

+ 1198916)

119892 (119905 119909) = 119891120575119885

119906 = 120575119885

(37)


119906 = minus120573

+ 2 (|| + |120590|23

)minus12

( + |120590|23 sign120590)

|| + 2 (|| + |120590|23

)12

(38)


0 lt 119870119898

le 119892 (119905 119909) le 119870119872

|ℎ (119905 119909)| le 119862

(39)




119892 (119905 119909) = 119891120575119885

= minus1

119903

119862120572

119884119876119878ref

119898

119898120575119885

119911119876119878ref119897ref

119869119885

cos (120579119872

minus 119902)

(40)


1198722 where 120588 =


=










120575119911



1003816100381610038161003816120579119872 minus 1199021003816100381610038161003816 lt

120587

2

997904rArr cos (120579119872

minus 119902) gt 0

(41)



119892 (119905 119909) gt 0 (42)



0 lt 119870119898

lt 119892 (119905 119909) (43)


0 lt 119870119898

lt 119892 (119905 119909) lt 119870119872

(44)

With (37) then

ℎ (119905 119909) =1

119903(1198911

+ 1198912

+ 1198913

+ 1198914

+ 1198915

+ 1198916) (45)


119902 119886119872 119886119879 and 119886



|ℎ (119905 119909)| le 119862 (46)


120575119885

= int 120575119885dt = int 119906

119894dt (47)


120590(4)

= ℎlowast

(119905 119909) + 119892lowast

(119905 119909) 119906119894= ℎlowast

(119905 119909) + 119892 (119905 119909) 119906119894

ℎlowast

(119905 119909) = ℎ (119905 119909) + 119892 (119905 119909) 120575119885

119892 (119905 119909)

=119898120575119885

1199111198762

1198782

119897119862120572

119884

119869119885

1198981199032[( 119902 minus 120579

119872) 119903 sin 120578

119872+ 119903 cos 120578

119872]

(48)


119872 119886119879 and 119886

119879 therefore




(119905 119909)| le 119862 (119862 gt 0)Because 120575


119885 119892lowast(119905 119909)


lt 119892lowast

(119905 119909) lt 119870119872

issatisfied


119894

119906119894= minus120573

Φ34

11987334

Φ34

=120590 + 3 [||

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

minus12

sdot [

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

11987334

=1003816100381610038161003816

120590

1003816100381610038161003816 + 3 [||

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

minus12

sdot

1003816100381610038161003816100381610038161003816

+ (|| + 05 |120590|34

)minus13

( + 05 |120590|34 sign120590)

1003816100381610038161003816100381610038161003816

(49)








valid1003816100381610038161003816

120590

1003816100381610038161003816 le 119862 + 120573119870119872

(50)






0

= V0

V0

= minus12058201198711(119899+1) 10038161003816100381610038161199110 minus 119891 (119905)

1003816100381610038161003816

119899(119899+1) sign (1199110

minus 119891 (119905))

+ 1199111

1

= V1

V1

= minus12058211198712(119899+1) 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

(119899minus1)119899 sign (1199111

minus V0) + 1199112

119899minus1

= V119899minus1

V119899minus1


11987112 1003816100381610038161003816119911119899minus1 minus V

119899minus2

1003816100381610038161003816

12 sign (119911119899minus1

minus V119899minus2

)

+ 119911119899

119899

= minus120582119899119871 sign (119911


)

(51)

and if 120582119894



0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(52)

where 1199113 1199112 1199111 and 119911


(4) 119902 119902 and 119902




119899119888

= minus119873 119902119881119872

119892 (53)





119899119888

= minus3100381610038161003816100381610038160

10038161003816100381610038161003816119902 + 120576

119902

10038161003816100381610038161199021003816100381610038161003816 + 120575

(54)


0is the approach






119868= 019 119870

120572= 3 and

119870120596



KIS

120596Z 120572 nY


modelServo


Step response

Time (s)0 02 04 06 08 1 12

0

02

04

06

08

1

Rise time (s) 0463


Bode diagram

Frequency (rads)




Am

plitu

de

Mag

nitu

de (d

B)Ph

ase (

deg)

0

minus180

minus360

minus54010410310210110010minus1

100

0

minus100

minus200

minus300





=

1205821

= 1205822

= 1205823

= 50 1199110

= 01 1199111

= 1199112

= 1199113




119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 0

(55)






119881119872








175

180

185

190

195

200

205

TargetHOSM-IGCPN

Y(k

m)

0 10 20 30 40 50 60X (km)


0

5

10

15

20

25

0 5 10 15 20Time (s)

HOSM-IGCPN



Miss

ile ac

cele

ratio

n (G

)

minus5




0

5

10

15

20

25

30

0 5 10 15Time (s)

Miss

ile ac

cele

ratio

n (G

)

minus5




0

1

2

0 5 10 15 20Time (s)

4 45 5

0

02

04

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 30)







HOSM-IGC(120573 = 10)

119881119905= 2000ms 115m 086 073m

119881119905= 3000ms 261m 117 106

119881119905= 4000ms 542m 156 134

0

1

2

0 5 10 15 20Time (s)

72 74 76 78

0

02

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)




119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

=

0 g 119905 lt 10 sec

5 g 119905 gt 10 sec

(56)






119881119872





15

17

19

21

23

25

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60X (km)







0

5

10

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus5

minus10


0

10

20

30

0 5 10 15 20Time (s)


tgo = 3 s (diverge)

tgo = 2 s (diverge)


minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 10)



0

10

20

30

0 5 10 15 20Time (s)


tgo = 1 s (diverge)

tgo = 2 s (diverge)


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

120573 = 30)HOSM-IGC (


0

10

20

30

0 5 10 15 20Time (s)

tgo = 1 s (diverge)




minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 50)






HOSM-IGC120573 = 30

HOSM-IGC120573 = 40

HOSM-IGC120573 = 50

OSMG

tgo = 1second 43539 3224 28936 36116

tgo = 2seconds 25424 18665 09322 35534

tgo = 3seconds 08124 08265 07538 11959

Time (s)0 5 10 15 20

0

10

20

30OSMG


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)









119881119872






119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 40 sin(120587119905

2)

(57)







0

05

1

15

2

25

3

35

4

45

5


OSMG

Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


HOSM-IGC (120573 = 40)

HOSM-IGC (120573 = 50)

HOSM-IGC (120573 = 30)



6 Conclusions



119885becomes


175

180

185

190

195

200

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60 0 20 40X (km)


0

10

20

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus10

minus20



0

10

20

30

0 5 10 15 20Time (s)

OSMGHOSM-IGC

minus10

minus20

minus30

minus40

Actu

ator

defl

ectio

n (d

eg)


0

50

100

150

200

15 16 17 18Time (s)


Miss

ile ac

cele

ratio

n (G

)

minus50


Appendices



0= 1205821

= 1205822

= 1205823

= 50


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0

100

200

300

400

0 02 04Time (s)

h(t)

z3

h(t) and z3

minus100

minus200

minus300

minus400

minus5000 02 04

Time (s)

g(t)

z2

g(t) and z2

minus100

minus200

minus300

minus400

f(t)

z1

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

F(t)

z0

F(t) and z0

minus0005

minus001



= 01 1199111

= 1199112

= 1199113


0

= V0

V0

= minus120582011987114 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

34 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987113 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

23 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987112 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

12 sign (1199112

minus V1) + 1199113

3

= minus1205823119871 sign (119911

3minus V2)

(A1)


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0 02 04Time (s)

0

200

400

0 02 04Time (s)

h(t) and z3

minus200

minus400

minus600

minus800

g(t) and z2

minus100

minus200

minus300

minus400

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

minus16

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0



int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +








0 02 04

0

0005

001

0015

002

0025

003

0035

004

Time (s)0 02 04

0

01

02

03

04

05

Time (s)

0 02 04

0

05

1

15

2

Time (s)0 02 04

0

05

1

15

2

25

3

35

4

Time (s)


minus04

minus05

f(t) and z1

minus2

minus15

minus1

minus05

minus1

minus05

minus01

minus02

minus03

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0




10038161003816100381610038161003816120590(119894)

10038161003816100381610038161003816le 120583120591119903minus119894

119894 = 0 119903 minus 1 (A2)








2has an obviously






0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)


= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825





Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2


1063 Kgsdotm2



See Table 7



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119889119881119872

119889119905= 0

119889120579119872

119889119905=

119884

119898119881119872

(11)

Denote that 119886119872119873


119886119879119873


120579119872

=119886119872119873

119881119872

120579119879

=119886119879119873

119881119879

(12)



119884 = 119862120572

119884120572119876119878ref + 119862

120575119885

119884120575119885119876119878ref (13)

119872119885

= 119898120575119885

119885120575119885119876119878ref119897 + 119898

120572

119885120572119876119878ref119897 (14)

where119862120572


of attack 119862120575119885




119885is orders of


119884 = 119862120572

119884120572119876119878ref (15)

Then

119886119872119873

=119884

119898=

119862120572

119884120572119876119878ref

119898 (16)


119889120579119872

119889119905=

119862120572

119884120572119876119878ref

119898119881119872

(17)

119889120596119885

119889119905=

119898120575119885

119885120575119885

119876119878ref119897 + 119898120572

119885120572119876119878ref119897

119869119885

(18)

119889120599

119889119905= 120596119885 (19)

120572 = 120599 minus 120579119872

(20)


( 119902 120579119872



119902 = minus2119903

119903sdot 119902 minus

119862120572

119884119876119878ref

119898119903cos (120579

119872minus 119902) sdot 120572

+119886119879119873

119903cos (120579

119879+ 119902)

119903 119902 = minus119881119872sin (120579119872

minus 119902) + 119881119879sin (120579119879

+ 119902)

120579119872

=1

119898119881119872

119862120572

119884119876119878ref sdot 120572

= 120596119885

minus1

119898119881119872

119862120572

119884119876119878ref sdot 120572

119885

=119898120575119885

119885119876119878ref119897

119869119885

sdot 120575119885

+119898120572

119885119876119878ref119897

119869119885

120572

(21)



=1

119903[minus2 119903 119902 minus 119886

119872119873cos (120579

119872minus 119902) + 119886

119879119873cos (120579

119879+ 119902)]

(22)



119902 =

1


119872119873( 119902 minus 120579

119872) sin (120579

119872minus 119902)

+ 119886119872119873

cos (120579119872

minus 119902)] minus [119886119879119873

( 119902 + 120579119879) sin (120579

119879+ 119902)

minus 119886119879119873

cos (120579119879

+ 119902)]

(23)



119886119872119873

=119862120572

119884119876119878ref

119898 =

119862120572

119884119876119878ref

119898( 120599 minus 120579

119872)

=119862120572

119884119876119878ref

119898(120596119885

minus 120579119872

)

(24)




119872as follows

119886119872119873

=119862120572

119884119876119878ref

119898(119885

minus119886119872

119881119872

)

=119862120572

119884119876119878ref

119898

119898120575119885

119885119876119878ref119897ref

119869119885

sdot 120575119885

+119862120572

119884119876119878ref

119898(

119898120572

119885119876119878ref119897ref

119869119885

120572 minus119886119872

119881119872

)

(25)


Thus

119902(4)

= 119891120575119885

sdot 120575119885

+1

119903(1198911

+ 1198912

+ 1198913

+ 1198914

+ 1198915

+ 1198916) (26)

where

119891120575119885

= minus1

119903

119862120572

119884119876119878ref cos 120578

119872

119898

119898120575119885

119885119876119878ref119897ref

119869119885

(27)

1198911

= (120596119885

minus 120579119872

) (minus2 120579119886119872119873

120572sin 120578119872

+ 119902119886119872119873

120572sin 120578119872

+ 120579119886119872119873

1205722cos 120578119872

) minus119886119872119873

cos 120578119872

120572

119898120572

119885120572119876119878ref119897ref

119869119885

(28)

1198912

= 120579119872

(( 120579119872

minus 119902) 119886119872119873

cos 120578119872

minus (120596119885

minus 120579119872

)119886119872119873

120572sin 120578119872

)

(29)

1198913

= 119902 (minus 120579119872

119886119872119873

cos 120578119872

+ 119902 (119886119872119873

cos 120578119872

minus 119886119879119873

cos 120578119879) minus 120579119879119886119879119873

cos 120578119879

+119886119872119873

120572(120596119885

minus 120579119872

) sin 120578119872

minus 119886119879sin 120578119879)

(30)

1198914

= 120579119879

(minus 119902119886119879cos 120578119879

minus 120579119879119886119879cos 120578119879

minus 119886119879sin 120578119879) (31)

1198915

= 119886119879

(minus 119902 sin 120578119879

minus 2 120579119879sin 120578119879) + 119886119879cos 120578119879 (32)

1198916

= 119902 (119886119872119873

sin 120578119872

minus 119886119879sin 120578119879


minus 2119903 sdot 119902

(33)

120578119872

= 120579119872

minus 119902

120578119879

= 120579119879

+ 119902

(34)



119885is 3



120590 = 119902 (35)



120590 = ℎ (119905 119909) + 119892 (119905 119909) 119906 (36)


ℎ (119905 119909) =1

119903(1198911

+ 1198912

+ 1198913

+ 1198914

+ 1198915

+ 1198916)

119892 (119905 119909) = 119891120575119885

119906 = 120575119885

(37)


119906 = minus120573

+ 2 (|| + |120590|23

)minus12

( + |120590|23 sign120590)

|| + 2 (|| + |120590|23

)12

(38)


0 lt 119870119898

le 119892 (119905 119909) le 119870119872

|ℎ (119905 119909)| le 119862

(39)




119892 (119905 119909) = 119891120575119885

= minus1

119903

119862120572

119884119876119878ref

119898

119898120575119885

119911119876119878ref119897ref

119869119885

cos (120579119872

minus 119902)

(40)


1198722 where 120588 =


=










120575119911



1003816100381610038161003816120579119872 minus 1199021003816100381610038161003816 lt

120587

2

997904rArr cos (120579119872

minus 119902) gt 0

(41)



119892 (119905 119909) gt 0 (42)



0 lt 119870119898

lt 119892 (119905 119909) (43)


0 lt 119870119898

lt 119892 (119905 119909) lt 119870119872

(44)

With (37) then

ℎ (119905 119909) =1

119903(1198911

+ 1198912

+ 1198913

+ 1198914

+ 1198915

+ 1198916) (45)


119902 119886119872 119886119879 and 119886



|ℎ (119905 119909)| le 119862 (46)


120575119885

= int 120575119885dt = int 119906

119894dt (47)


120590(4)

= ℎlowast

(119905 119909) + 119892lowast

(119905 119909) 119906119894= ℎlowast

(119905 119909) + 119892 (119905 119909) 119906119894

ℎlowast

(119905 119909) = ℎ (119905 119909) + 119892 (119905 119909) 120575119885

119892 (119905 119909)

=119898120575119885

1199111198762

1198782

119897119862120572

119884

119869119885

1198981199032[( 119902 minus 120579

119872) 119903 sin 120578

119872+ 119903 cos 120578

119872]

(48)


119872 119886119879 and 119886

119879 therefore




(119905 119909)| le 119862 (119862 gt 0)Because 120575


119885 119892lowast(119905 119909)


lt 119892lowast

(119905 119909) lt 119870119872

issatisfied


119894

119906119894= minus120573

Φ34

11987334

Φ34

=120590 + 3 [||

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

minus12

sdot [

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

11987334

=1003816100381610038161003816

120590

1003816100381610038161003816 + 3 [||

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

minus12

sdot

1003816100381610038161003816100381610038161003816

+ (|| + 05 |120590|34

)minus13

( + 05 |120590|34 sign120590)

1003816100381610038161003816100381610038161003816

(49)








valid1003816100381610038161003816

120590

1003816100381610038161003816 le 119862 + 120573119870119872

(50)






0

= V0

V0

= minus12058201198711(119899+1) 10038161003816100381610038161199110 minus 119891 (119905)

1003816100381610038161003816

119899(119899+1) sign (1199110

minus 119891 (119905))

+ 1199111

1

= V1

V1

= minus12058211198712(119899+1) 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

(119899minus1)119899 sign (1199111

minus V0) + 1199112

119899minus1

= V119899minus1

V119899minus1


11987112 1003816100381610038161003816119911119899minus1 minus V

119899minus2

1003816100381610038161003816

12 sign (119911119899minus1

minus V119899minus2

)

+ 119911119899

119899

= minus120582119899119871 sign (119911


)

(51)

and if 120582119894



0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(52)

where 1199113 1199112 1199111 and 119911


(4) 119902 119902 and 119902




119899119888

= minus119873 119902119881119872

119892 (53)





119899119888

= minus3100381610038161003816100381610038160

10038161003816100381610038161003816119902 + 120576

119902

10038161003816100381610038161199021003816100381610038161003816 + 120575

(54)


0is the approach






119868= 019 119870

120572= 3 and

119870120596



KIS

120596Z 120572 nY


modelServo


Step response

Time (s)0 02 04 06 08 1 12

0

02

04

06

08

1

Rise time (s) 0463


Bode diagram

Frequency (rads)




Am

plitu

de

Mag

nitu

de (d

B)Ph

ase (

deg)

0

minus180

minus360

minus54010410310210110010minus1

100

0

minus100

minus200

minus300





=

1205821

= 1205822

= 1205823

= 50 1199110

= 01 1199111

= 1199112

= 1199113




119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 0

(55)






119881119872








175

180

185

190

195

200

205

TargetHOSM-IGCPN

Y(k

m)

0 10 20 30 40 50 60X (km)


0

5

10

15

20

25

0 5 10 15 20Time (s)

HOSM-IGCPN



Miss

ile ac

cele

ratio

n (G

)

minus5




0

5

10

15

20

25

30

0 5 10 15Time (s)

Miss

ile ac

cele

ratio

n (G

)

minus5




0

1

2

0 5 10 15 20Time (s)

4 45 5

0

02

04

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 30)







HOSM-IGC(120573 = 10)

119881119905= 2000ms 115m 086 073m

119881119905= 3000ms 261m 117 106

119881119905= 4000ms 542m 156 134

0

1

2

0 5 10 15 20Time (s)

72 74 76 78

0

02

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)




119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

=

0 g 119905 lt 10 sec

5 g 119905 gt 10 sec

(56)






119881119872





15

17

19

21

23

25

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60X (km)







0

5

10

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus5

minus10


0

10

20

30

0 5 10 15 20Time (s)


tgo = 3 s (diverge)

tgo = 2 s (diverge)


minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 10)



0

10

20

30

0 5 10 15 20Time (s)


tgo = 1 s (diverge)

tgo = 2 s (diverge)


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

120573 = 30)HOSM-IGC (


0

10

20

30

0 5 10 15 20Time (s)

tgo = 1 s (diverge)




minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 50)






HOSM-IGC120573 = 30

HOSM-IGC120573 = 40

HOSM-IGC120573 = 50

OSMG

tgo = 1second 43539 3224 28936 36116

tgo = 2seconds 25424 18665 09322 35534

tgo = 3seconds 08124 08265 07538 11959

Time (s)0 5 10 15 20

0

10

20

30OSMG


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)









119881119872






119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 40 sin(120587119905

2)

(57)







0

05

1

15

2

25

3

35

4

45

5


OSMG

Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


HOSM-IGC (120573 = 40)

HOSM-IGC (120573 = 50)

HOSM-IGC (120573 = 30)



6 Conclusions



119885becomes


175

180

185

190

195

200

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60 0 20 40X (km)


0

10

20

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus10

minus20



0

10

20

30

0 5 10 15 20Time (s)

OSMGHOSM-IGC

minus10

minus20

minus30

minus40

Actu

ator

defl

ectio

n (d

eg)


0

50

100

150

200

15 16 17 18Time (s)


Miss

ile ac

cele

ratio

n (G

)

minus50


Appendices



0= 1205821

= 1205822

= 1205823

= 50


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0

100

200

300

400

0 02 04Time (s)

h(t)

z3

h(t) and z3

minus100

minus200

minus300

minus400

minus5000 02 04

Time (s)

g(t)

z2

g(t) and z2

minus100

minus200

minus300

minus400

f(t)

z1

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

F(t)

z0

F(t) and z0

minus0005

minus001



= 01 1199111

= 1199112

= 1199113


0

= V0

V0

= minus120582011987114 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

34 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987113 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

23 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987112 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

12 sign (1199112

minus V1) + 1199113

3

= minus1205823119871 sign (119911

3minus V2)

(A1)


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0 02 04Time (s)

0

200

400

0 02 04Time (s)

h(t) and z3

minus200

minus400

minus600

minus800

g(t) and z2

minus100

minus200

minus300

minus400

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

minus16

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0



int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +








0 02 04

0

0005

001

0015

002

0025

003

0035

004

Time (s)0 02 04

0

01

02

03

04

05

Time (s)

0 02 04

0

05

1

15

2

Time (s)0 02 04

0

05

1

15

2

25

3

35

4

Time (s)


minus04

minus05

f(t) and z1

minus2

minus15

minus1

minus05

minus1

minus05

minus01

minus02

minus03

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0




10038161003816100381610038161003816120590(119894)

10038161003816100381610038161003816le 120583120591119903minus119894

119894 = 0 119903 minus 1 (A2)








2has an obviously






0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)


= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825





Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2


1063 Kgsdotm2



See Table 7



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Thus

119902(4)

= 119891120575119885

sdot 120575119885

+1

119903(1198911

+ 1198912

+ 1198913

+ 1198914

+ 1198915

+ 1198916) (26)

where

119891120575119885

= minus1

119903

119862120572

119884119876119878ref cos 120578

119872

119898

119898120575119885

119885119876119878ref119897ref

119869119885

(27)

1198911

= (120596119885

minus 120579119872

) (minus2 120579119886119872119873

120572sin 120578119872

+ 119902119886119872119873

120572sin 120578119872

+ 120579119886119872119873

1205722cos 120578119872

) minus119886119872119873

cos 120578119872

120572

119898120572

119885120572119876119878ref119897ref

119869119885

(28)

1198912

= 120579119872

(( 120579119872

minus 119902) 119886119872119873

cos 120578119872

minus (120596119885

minus 120579119872

)119886119872119873

120572sin 120578119872

)

(29)

1198913

= 119902 (minus 120579119872

119886119872119873

cos 120578119872

+ 119902 (119886119872119873

cos 120578119872

minus 119886119879119873

cos 120578119879) minus 120579119879119886119879119873

cos 120578119879

+119886119872119873

120572(120596119885

minus 120579119872

) sin 120578119872

minus 119886119879sin 120578119879)

(30)

1198914

= 120579119879

(minus 119902119886119879cos 120578119879

minus 120579119879119886119879cos 120578119879

minus 119886119879sin 120578119879) (31)

1198915

= 119886119879

(minus 119902 sin 120578119879

minus 2 120579119879sin 120578119879) + 119886119879cos 120578119879 (32)

1198916

= 119902 (119886119872119873

sin 120578119872

minus 119886119879sin 120578119879


minus 2119903 sdot 119902

(33)

120578119872

= 120579119872

minus 119902

120578119879

= 120579119879

+ 119902

(34)



119885is 3



120590 = 119902 (35)



120590 = ℎ (119905 119909) + 119892 (119905 119909) 119906 (36)


ℎ (119905 119909) =1

119903(1198911

+ 1198912

+ 1198913

+ 1198914

+ 1198915

+ 1198916)

119892 (119905 119909) = 119891120575119885

119906 = 120575119885

(37)


119906 = minus120573

+ 2 (|| + |120590|23

)minus12

( + |120590|23 sign120590)

|| + 2 (|| + |120590|23

)12

(38)


0 lt 119870119898

le 119892 (119905 119909) le 119870119872

|ℎ (119905 119909)| le 119862

(39)




119892 (119905 119909) = 119891120575119885

= minus1

119903

119862120572

119884119876119878ref

119898

119898120575119885

119911119876119878ref119897ref

119869119885

cos (120579119872

minus 119902)

(40)


1198722 where 120588 =


=










120575119911



1003816100381610038161003816120579119872 minus 1199021003816100381610038161003816 lt

120587

2

997904rArr cos (120579119872

minus 119902) gt 0

(41)



119892 (119905 119909) gt 0 (42)



0 lt 119870119898

lt 119892 (119905 119909) (43)


0 lt 119870119898

lt 119892 (119905 119909) lt 119870119872

(44)

With (37) then

ℎ (119905 119909) =1

119903(1198911

+ 1198912

+ 1198913

+ 1198914

+ 1198915

+ 1198916) (45)


119902 119886119872 119886119879 and 119886



|ℎ (119905 119909)| le 119862 (46)


120575119885

= int 120575119885dt = int 119906

119894dt (47)


120590(4)

= ℎlowast

(119905 119909) + 119892lowast

(119905 119909) 119906119894= ℎlowast

(119905 119909) + 119892 (119905 119909) 119906119894

ℎlowast

(119905 119909) = ℎ (119905 119909) + 119892 (119905 119909) 120575119885

119892 (119905 119909)

=119898120575119885

1199111198762

1198782

119897119862120572

119884

119869119885

1198981199032[( 119902 minus 120579

119872) 119903 sin 120578

119872+ 119903 cos 120578

119872]

(48)


119872 119886119879 and 119886

119879 therefore




(119905 119909)| le 119862 (119862 gt 0)Because 120575


119885 119892lowast(119905 119909)


lt 119892lowast

(119905 119909) lt 119870119872

issatisfied


119894

119906119894= minus120573

Φ34

11987334

Φ34

=120590 + 3 [||

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

minus12

sdot [

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

11987334

=1003816100381610038161003816

120590

1003816100381610038161003816 + 3 [||

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

minus12

sdot

1003816100381610038161003816100381610038161003816

+ (|| + 05 |120590|34

)minus13

( + 05 |120590|34 sign120590)

1003816100381610038161003816100381610038161003816

(49)








valid1003816100381610038161003816

120590

1003816100381610038161003816 le 119862 + 120573119870119872

(50)






0

= V0

V0

= minus12058201198711(119899+1) 10038161003816100381610038161199110 minus 119891 (119905)

1003816100381610038161003816

119899(119899+1) sign (1199110

minus 119891 (119905))

+ 1199111

1

= V1

V1

= minus12058211198712(119899+1) 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

(119899minus1)119899 sign (1199111

minus V0) + 1199112

119899minus1

= V119899minus1

V119899minus1


11987112 1003816100381610038161003816119911119899minus1 minus V

119899minus2

1003816100381610038161003816

12 sign (119911119899minus1

minus V119899minus2

)

+ 119911119899

119899

= minus120582119899119871 sign (119911


)

(51)

and if 120582119894



0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(52)

where 1199113 1199112 1199111 and 119911


(4) 119902 119902 and 119902




119899119888

= minus119873 119902119881119872

119892 (53)





119899119888

= minus3100381610038161003816100381610038160

10038161003816100381610038161003816119902 + 120576

119902

10038161003816100381610038161199021003816100381610038161003816 + 120575

(54)


0is the approach






119868= 019 119870

120572= 3 and

119870120596



KIS

120596Z 120572 nY


modelServo


Step response

Time (s)0 02 04 06 08 1 12

0

02

04

06

08

1

Rise time (s) 0463


Bode diagram

Frequency (rads)




Am

plitu

de

Mag

nitu

de (d

B)Ph

ase (

deg)

0

minus180

minus360

minus54010410310210110010minus1

100

0

minus100

minus200

minus300





=

1205821

= 1205822

= 1205823

= 50 1199110

= 01 1199111

= 1199112

= 1199113




119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 0

(55)






119881119872








175

180

185

190

195

200

205

TargetHOSM-IGCPN

Y(k

m)

0 10 20 30 40 50 60X (km)


0

5

10

15

20

25

0 5 10 15 20Time (s)

HOSM-IGCPN



Miss

ile ac

cele

ratio

n (G

)

minus5




0

5

10

15

20

25

30

0 5 10 15Time (s)

Miss

ile ac

cele

ratio

n (G

)

minus5




0

1

2

0 5 10 15 20Time (s)

4 45 5

0

02

04

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 30)







HOSM-IGC(120573 = 10)

119881119905= 2000ms 115m 086 073m

119881119905= 3000ms 261m 117 106

119881119905= 4000ms 542m 156 134

0

1

2

0 5 10 15 20Time (s)

72 74 76 78

0

02

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)




119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

=

0 g 119905 lt 10 sec

5 g 119905 gt 10 sec

(56)






119881119872





15

17

19

21

23

25

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60X (km)







0

5

10

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus5

minus10


0

10

20

30

0 5 10 15 20Time (s)


tgo = 3 s (diverge)

tgo = 2 s (diverge)


minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 10)



0

10

20

30

0 5 10 15 20Time (s)


tgo = 1 s (diverge)

tgo = 2 s (diverge)


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

120573 = 30)HOSM-IGC (


0

10

20

30

0 5 10 15 20Time (s)

tgo = 1 s (diverge)




minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 50)






HOSM-IGC120573 = 30

HOSM-IGC120573 = 40

HOSM-IGC120573 = 50

OSMG

tgo = 1second 43539 3224 28936 36116

tgo = 2seconds 25424 18665 09322 35534

tgo = 3seconds 08124 08265 07538 11959

Time (s)0 5 10 15 20

0

10

20

30OSMG


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)









119881119872






119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 40 sin(120587119905

2)

(57)







0

05

1

15

2

25

3

35

4

45

5


OSMG

Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


HOSM-IGC (120573 = 40)

HOSM-IGC (120573 = 50)

HOSM-IGC (120573 = 30)



6 Conclusions



119885becomes


175

180

185

190

195

200

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60 0 20 40X (km)


0

10

20

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus10

minus20



0

10

20

30

0 5 10 15 20Time (s)

OSMGHOSM-IGC

minus10

minus20

minus30

minus40

Actu

ator

defl

ectio

n (d

eg)


0

50

100

150

200

15 16 17 18Time (s)


Miss

ile ac

cele

ratio

n (G

)

minus50


Appendices



0= 1205821

= 1205822

= 1205823

= 50


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0

100

200

300

400

0 02 04Time (s)

h(t)

z3

h(t) and z3

minus100

minus200

minus300

minus400

minus5000 02 04

Time (s)

g(t)

z2

g(t) and z2

minus100

minus200

minus300

minus400

f(t)

z1

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

F(t)

z0

F(t) and z0

minus0005

minus001



= 01 1199111

= 1199112

= 1199113


0

= V0

V0

= minus120582011987114 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

34 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987113 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

23 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987112 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

12 sign (1199112

minus V1) + 1199113

3

= minus1205823119871 sign (119911

3minus V2)

(A1)


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0 02 04Time (s)

0

200

400

0 02 04Time (s)

h(t) and z3

minus200

minus400

minus600

minus800

g(t) and z2

minus100

minus200

minus300

minus400

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

minus16

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0



int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +








0 02 04

0

0005

001

0015

002

0025

003

0035

004

Time (s)0 02 04

0

01

02

03

04

05

Time (s)

0 02 04

0

05

1

15

2

Time (s)0 02 04

0

05

1

15

2

25

3

35

4

Time (s)


minus04

minus05

f(t) and z1

minus2

minus15

minus1

minus05

minus1

minus05

minus01

minus02

minus03

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0




10038161003816100381610038161003816120590(119894)

10038161003816100381610038161003816le 120583120591119903minus119894

119894 = 0 119903 minus 1 (A2)








2has an obviously






0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)


= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825





Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2


1063 Kgsdotm2



See Table 7



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119892 (119905 119909) gt 0 (42)



0 lt 119870119898

lt 119892 (119905 119909) (43)


0 lt 119870119898

lt 119892 (119905 119909) lt 119870119872

(44)

With (37) then

ℎ (119905 119909) =1

119903(1198911

+ 1198912

+ 1198913

+ 1198914

+ 1198915

+ 1198916) (45)


119902 119886119872 119886119879 and 119886



|ℎ (119905 119909)| le 119862 (46)


120575119885

= int 120575119885dt = int 119906

119894dt (47)


120590(4)

= ℎlowast

(119905 119909) + 119892lowast

(119905 119909) 119906119894= ℎlowast

(119905 119909) + 119892 (119905 119909) 119906119894

ℎlowast

(119905 119909) = ℎ (119905 119909) + 119892 (119905 119909) 120575119885

119892 (119905 119909)

=119898120575119885

1199111198762

1198782

119897119862120572

119884

119869119885

1198981199032[( 119902 minus 120579

119872) 119903 sin 120578

119872+ 119903 cos 120578

119872]

(48)


119872 119886119879 and 119886

119879 therefore




(119905 119909)| le 119862 (119862 gt 0)Because 120575


119885 119892lowast(119905 119909)


lt 119892lowast

(119905 119909) lt 119870119872

issatisfied


119894

119906119894= minus120573

Φ34

11987334

Φ34

=120590 + 3 [||

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

minus12

sdot [

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

11987334

=1003816100381610038161003816

120590

1003816100381610038161003816 + 3 [||

+ (|| + 05 |120590|34

)minus13 10038161003816100381610038161003816

+ 05 |120590|34 sign120590

10038161003816100381610038161003816]

minus12

sdot

1003816100381610038161003816100381610038161003816

+ (|| + 05 |120590|34

)minus13

( + 05 |120590|34 sign120590)

1003816100381610038161003816100381610038161003816

(49)








valid1003816100381610038161003816

120590

1003816100381610038161003816 le 119862 + 120573119870119872

(50)






0

= V0

V0

= minus12058201198711(119899+1) 10038161003816100381610038161199110 minus 119891 (119905)

1003816100381610038161003816

119899(119899+1) sign (1199110

minus 119891 (119905))

+ 1199111

1

= V1

V1

= minus12058211198712(119899+1) 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

(119899minus1)119899 sign (1199111

minus V0) + 1199112

119899minus1

= V119899minus1

V119899minus1


11987112 1003816100381610038161003816119911119899minus1 minus V

119899minus2

1003816100381610038161003816

12 sign (119911119899minus1

minus V119899minus2

)

+ 119911119899

119899

= minus120582119899119871 sign (119911


)

(51)

and if 120582119894



0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(52)

where 1199113 1199112 1199111 and 119911


(4) 119902 119902 and 119902




119899119888

= minus119873 119902119881119872

119892 (53)





119899119888

= minus3100381610038161003816100381610038160

10038161003816100381610038161003816119902 + 120576

119902

10038161003816100381610038161199021003816100381610038161003816 + 120575

(54)


0is the approach






119868= 019 119870

120572= 3 and

119870120596



KIS

120596Z 120572 nY


modelServo


Step response

Time (s)0 02 04 06 08 1 12

0

02

04

06

08

1

Rise time (s) 0463


Bode diagram

Frequency (rads)




Am

plitu

de

Mag

nitu

de (d

B)Ph

ase (

deg)

0

minus180

minus360

minus54010410310210110010minus1

100

0

minus100

minus200

minus300





=

1205821

= 1205822

= 1205823

= 50 1199110

= 01 1199111

= 1199112

= 1199113




119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 0

(55)






119881119872








175

180

185

190

195

200

205

TargetHOSM-IGCPN

Y(k

m)

0 10 20 30 40 50 60X (km)


0

5

10

15

20

25

0 5 10 15 20Time (s)

HOSM-IGCPN



Miss

ile ac

cele

ratio

n (G

)

minus5




0

5

10

15

20

25

30

0 5 10 15Time (s)

Miss

ile ac

cele

ratio

n (G

)

minus5




0

1

2

0 5 10 15 20Time (s)

4 45 5

0

02

04

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 30)







HOSM-IGC(120573 = 10)

119881119905= 2000ms 115m 086 073m

119881119905= 3000ms 261m 117 106

119881119905= 4000ms 542m 156 134

0

1

2

0 5 10 15 20Time (s)

72 74 76 78

0

02

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)




119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

=

0 g 119905 lt 10 sec

5 g 119905 gt 10 sec

(56)






119881119872





15

17

19

21

23

25

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60X (km)







0

5

10

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus5

minus10


0

10

20

30

0 5 10 15 20Time (s)


tgo = 3 s (diverge)

tgo = 2 s (diverge)


minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 10)



0

10

20

30

0 5 10 15 20Time (s)


tgo = 1 s (diverge)

tgo = 2 s (diverge)


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

120573 = 30)HOSM-IGC (


0

10

20

30

0 5 10 15 20Time (s)

tgo = 1 s (diverge)




minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 50)






HOSM-IGC120573 = 30

HOSM-IGC120573 = 40

HOSM-IGC120573 = 50

OSMG

tgo = 1second 43539 3224 28936 36116

tgo = 2seconds 25424 18665 09322 35534

tgo = 3seconds 08124 08265 07538 11959

Time (s)0 5 10 15 20

0

10

20

30OSMG


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)









119881119872






119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 40 sin(120587119905

2)

(57)







0

05

1

15

2

25

3

35

4

45

5


OSMG

Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


HOSM-IGC (120573 = 40)

HOSM-IGC (120573 = 50)

HOSM-IGC (120573 = 30)



6 Conclusions



119885becomes


175

180

185

190

195

200

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60 0 20 40X (km)


0

10

20

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus10

minus20



0

10

20

30

0 5 10 15 20Time (s)

OSMGHOSM-IGC

minus10

minus20

minus30

minus40

Actu

ator

defl

ectio

n (d

eg)


0

50

100

150

200

15 16 17 18Time (s)


Miss

ile ac

cele

ratio

n (G

)

minus50


Appendices



0= 1205821

= 1205822

= 1205823

= 50


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0

100

200

300

400

0 02 04Time (s)

h(t)

z3

h(t) and z3

minus100

minus200

minus300

minus400

minus5000 02 04

Time (s)

g(t)

z2

g(t) and z2

minus100

minus200

minus300

minus400

f(t)

z1

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

F(t)

z0

F(t) and z0

minus0005

minus001



= 01 1199111

= 1199112

= 1199113


0

= V0

V0

= minus120582011987114 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

34 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987113 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

23 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987112 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

12 sign (1199112

minus V1) + 1199113

3

= minus1205823119871 sign (119911

3minus V2)

(A1)


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0 02 04Time (s)

0

200

400

0 02 04Time (s)

h(t) and z3

minus200

minus400

minus600

minus800

g(t) and z2

minus100

minus200

minus300

minus400

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

minus16

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0



int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +








0 02 04

0

0005

001

0015

002

0025

003

0035

004

Time (s)0 02 04

0

01

02

03

04

05

Time (s)

0 02 04

0

05

1

15

2

Time (s)0 02 04

0

05

1

15

2

25

3

35

4

Time (s)


minus04

minus05

f(t) and z1

minus2

minus15

minus1

minus05

minus1

minus05

minus01

minus02

minus03

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0




10038161003816100381610038161003816120590(119894)

10038161003816100381610038161003816le 120583120591119903minus119894

119894 = 0 119903 minus 1 (A2)








2has an obviously






0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)


= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825





Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2


1063 Kgsdotm2



See Table 7



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0

= V0

V0

= minus12058201198711(119899+1) 10038161003816100381610038161199110 minus 119891 (119905)

1003816100381610038161003816

119899(119899+1) sign (1199110

minus 119891 (119905))

+ 1199111

1

= V1

V1

= minus12058211198712(119899+1) 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

(119899minus1)119899 sign (1199111

minus V0) + 1199112

119899minus1

= V119899minus1

V119899minus1


11987112 1003816100381610038161003816119911119899minus1 minus V

119899minus2

1003816100381610038161003816

12 sign (119911119899minus1

minus V119899minus2

)

+ 119911119899

119899

= minus120582119899119871 sign (119911


)

(51)

and if 120582119894



0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(52)

where 1199113 1199112 1199111 and 119911


(4) 119902 119902 and 119902




119899119888

= minus119873 119902119881119872

119892 (53)





119899119888

= minus3100381610038161003816100381610038160

10038161003816100381610038161003816119902 + 120576

119902

10038161003816100381610038161199021003816100381610038161003816 + 120575

(54)


0is the approach






119868= 019 119870

120572= 3 and

119870120596



KIS

120596Z 120572 nY


modelServo


Step response

Time (s)0 02 04 06 08 1 12

0

02

04

06

08

1

Rise time (s) 0463


Bode diagram

Frequency (rads)




Am

plitu

de

Mag

nitu

de (d

B)Ph

ase (

deg)

0

minus180

minus360

minus54010410310210110010minus1

100

0

minus100

minus200

minus300





=

1205821

= 1205822

= 1205823

= 50 1199110

= 01 1199111

= 1199112

= 1199113




119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 0

(55)






119881119872








175

180

185

190

195

200

205

TargetHOSM-IGCPN

Y(k

m)

0 10 20 30 40 50 60X (km)


0

5

10

15

20

25

0 5 10 15 20Time (s)

HOSM-IGCPN



Miss

ile ac

cele

ratio

n (G

)

minus5




0

5

10

15

20

25

30

0 5 10 15Time (s)

Miss

ile ac

cele

ratio

n (G

)

minus5




0

1

2

0 5 10 15 20Time (s)

4 45 5

0

02

04

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 30)







HOSM-IGC(120573 = 10)

119881119905= 2000ms 115m 086 073m

119881119905= 3000ms 261m 117 106

119881119905= 4000ms 542m 156 134

0

1

2

0 5 10 15 20Time (s)

72 74 76 78

0

02

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)




119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

=

0 g 119905 lt 10 sec

5 g 119905 gt 10 sec

(56)






119881119872





15

17

19

21

23

25

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60X (km)







0

5

10

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus5

minus10


0

10

20

30

0 5 10 15 20Time (s)


tgo = 3 s (diverge)

tgo = 2 s (diverge)


minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 10)



0

10

20

30

0 5 10 15 20Time (s)


tgo = 1 s (diverge)

tgo = 2 s (diverge)


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

120573 = 30)HOSM-IGC (


0

10

20

30

0 5 10 15 20Time (s)

tgo = 1 s (diverge)




minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 50)






HOSM-IGC120573 = 30

HOSM-IGC120573 = 40

HOSM-IGC120573 = 50

OSMG

tgo = 1second 43539 3224 28936 36116

tgo = 2seconds 25424 18665 09322 35534

tgo = 3seconds 08124 08265 07538 11959

Time (s)0 5 10 15 20

0

10

20

30OSMG


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)









119881119872






119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 40 sin(120587119905

2)

(57)







0

05

1

15

2

25

3

35

4

45

5


OSMG

Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


HOSM-IGC (120573 = 40)

HOSM-IGC (120573 = 50)

HOSM-IGC (120573 = 30)



6 Conclusions



119885becomes


175

180

185

190

195

200

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60 0 20 40X (km)


0

10

20

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus10

minus20



0

10

20

30

0 5 10 15 20Time (s)

OSMGHOSM-IGC

minus10

minus20

minus30

minus40

Actu

ator

defl

ectio

n (d

eg)


0

50

100

150

200

15 16 17 18Time (s)


Miss

ile ac

cele

ratio

n (G

)

minus50


Appendices



0= 1205821

= 1205822

= 1205823

= 50


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0

100

200

300

400

0 02 04Time (s)

h(t)

z3

h(t) and z3

minus100

minus200

minus300

minus400

minus5000 02 04

Time (s)

g(t)

z2

g(t) and z2

minus100

minus200

minus300

minus400

f(t)

z1

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

F(t)

z0

F(t) and z0

minus0005

minus001



= 01 1199111

= 1199112

= 1199113


0

= V0

V0

= minus120582011987114 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

34 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987113 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

23 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987112 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

12 sign (1199112

minus V1) + 1199113

3

= minus1205823119871 sign (119911

3minus V2)

(A1)


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0 02 04Time (s)

0

200

400

0 02 04Time (s)

h(t) and z3

minus200

minus400

minus600

minus800

g(t) and z2

minus100

minus200

minus300

minus400

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

minus16

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0



int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +








0 02 04

0

0005

001

0015

002

0025

003

0035

004

Time (s)0 02 04

0

01

02

03

04

05

Time (s)

0 02 04

0

05

1

15

2

Time (s)0 02 04

0

05

1

15

2

25

3

35

4

Time (s)


minus04

minus05

f(t) and z1

minus2

minus15

minus1

minus05

minus1

minus05

minus01

minus02

minus03

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0




10038161003816100381610038161003816120590(119894)

10038161003816100381610038161003816le 120583120591119903minus119894

119894 = 0 119903 minus 1 (A2)








2has an obviously






0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)


= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825





Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2


1063 Kgsdotm2



See Table 7



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










KIS

120596Z 120572 nY


modelServo


Step response

Time (s)0 02 04 06 08 1 12

0

02

04

06

08

1

Rise time (s) 0463


Bode diagram

Frequency (rads)




Am

plitu

de

Mag

nitu

de (d

B)Ph

ase (

deg)

0

minus180

minus360

minus54010410310210110010minus1

100

0

minus100

minus200

minus300





=

1205821

= 1205822

= 1205823

= 50 1199110

= 01 1199111

= 1199112

= 1199113




119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 0

(55)






119881119872








175

180

185

190

195

200

205

TargetHOSM-IGCPN

Y(k

m)

0 10 20 30 40 50 60X (km)


0

5

10

15

20

25

0 5 10 15 20Time (s)

HOSM-IGCPN



Miss

ile ac

cele

ratio

n (G

)

minus5




0

5

10

15

20

25

30

0 5 10 15Time (s)

Miss

ile ac

cele

ratio

n (G

)

minus5




0

1

2

0 5 10 15 20Time (s)

4 45 5

0

02

04

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 30)







HOSM-IGC(120573 = 10)

119881119905= 2000ms 115m 086 073m

119881119905= 3000ms 261m 117 106

119881119905= 4000ms 542m 156 134

0

1

2

0 5 10 15 20Time (s)

72 74 76 78

0

02

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)




119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

=

0 g 119905 lt 10 sec

5 g 119905 gt 10 sec

(56)






119881119872





15

17

19

21

23

25

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60X (km)







0

5

10

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus5

minus10


0

10

20

30

0 5 10 15 20Time (s)


tgo = 3 s (diverge)

tgo = 2 s (diverge)


minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 10)



0

10

20

30

0 5 10 15 20Time (s)


tgo = 1 s (diverge)

tgo = 2 s (diverge)


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

120573 = 30)HOSM-IGC (


0

10

20

30

0 5 10 15 20Time (s)

tgo = 1 s (diverge)




minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 50)






HOSM-IGC120573 = 30

HOSM-IGC120573 = 40

HOSM-IGC120573 = 50

OSMG

tgo = 1second 43539 3224 28936 36116

tgo = 2seconds 25424 18665 09322 35534

tgo = 3seconds 08124 08265 07538 11959

Time (s)0 5 10 15 20

0

10

20

30OSMG


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)









119881119872






119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 40 sin(120587119905

2)

(57)







0

05

1

15

2

25

3

35

4

45

5


OSMG

Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


HOSM-IGC (120573 = 40)

HOSM-IGC (120573 = 50)

HOSM-IGC (120573 = 30)



6 Conclusions



119885becomes


175

180

185

190

195

200

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60 0 20 40X (km)


0

10

20

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus10

minus20



0

10

20

30

0 5 10 15 20Time (s)

OSMGHOSM-IGC

minus10

minus20

minus30

minus40

Actu

ator

defl

ectio

n (d

eg)


0

50

100

150

200

15 16 17 18Time (s)


Miss

ile ac

cele

ratio

n (G

)

minus50


Appendices



0= 1205821

= 1205822

= 1205823

= 50


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0

100

200

300

400

0 02 04Time (s)

h(t)

z3

h(t) and z3

minus100

minus200

minus300

minus400

minus5000 02 04

Time (s)

g(t)

z2

g(t) and z2

minus100

minus200

minus300

minus400

f(t)

z1

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

F(t)

z0

F(t) and z0

minus0005

minus001



= 01 1199111

= 1199112

= 1199113


0

= V0

V0

= minus120582011987114 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

34 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987113 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

23 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987112 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

12 sign (1199112

minus V1) + 1199113

3

= minus1205823119871 sign (119911

3minus V2)

(A1)


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0 02 04Time (s)

0

200

400

0 02 04Time (s)

h(t) and z3

minus200

minus400

minus600

minus800

g(t) and z2

minus100

minus200

minus300

minus400

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

minus16

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0



int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +








0 02 04

0

0005

001

0015

002

0025

003

0035

004

Time (s)0 02 04

0

01

02

03

04

05

Time (s)

0 02 04

0

05

1

15

2

Time (s)0 02 04

0

05

1

15

2

25

3

35

4

Time (s)


minus04

minus05

f(t) and z1

minus2

minus15

minus1

minus05

minus1

minus05

minus01

minus02

minus03

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0




10038161003816100381610038161003816120590(119894)

10038161003816100381610038161003816le 120583120591119903minus119894

119894 = 0 119903 minus 1 (A2)








2has an obviously






0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)


= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825





Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2


1063 Kgsdotm2



See Table 7



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










175

180

185

190

195

200

205

TargetHOSM-IGCPN

Y(k

m)

0 10 20 30 40 50 60X (km)


0

5

10

15

20

25

0 5 10 15 20Time (s)

HOSM-IGCPN



Miss

ile ac

cele

ratio

n (G

)

minus5




0

5

10

15

20

25

30

0 5 10 15Time (s)

Miss

ile ac

cele

ratio

n (G

)

minus5




0

1

2

0 5 10 15 20Time (s)

4 45 5

0

02

04

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 30)







HOSM-IGC(120573 = 10)

119881119905= 2000ms 115m 086 073m

119881119905= 3000ms 261m 117 106

119881119905= 4000ms 542m 156 134

0

1

2

0 5 10 15 20Time (s)

72 74 76 78

0

02

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)




119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

=

0 g 119905 lt 10 sec

5 g 119905 gt 10 sec

(56)






119881119872





15

17

19

21

23

25

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60X (km)







0

5

10

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus5

minus10


0

10

20

30

0 5 10 15 20Time (s)


tgo = 3 s (diverge)

tgo = 2 s (diverge)


minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 10)



0

10

20

30

0 5 10 15 20Time (s)


tgo = 1 s (diverge)

tgo = 2 s (diverge)


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

120573 = 30)HOSM-IGC (


0

10

20

30

0 5 10 15 20Time (s)

tgo = 1 s (diverge)




minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 50)






HOSM-IGC120573 = 30

HOSM-IGC120573 = 40

HOSM-IGC120573 = 50

OSMG

tgo = 1second 43539 3224 28936 36116

tgo = 2seconds 25424 18665 09322 35534

tgo = 3seconds 08124 08265 07538 11959

Time (s)0 5 10 15 20

0

10

20

30OSMG


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)









119881119872






119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 40 sin(120587119905

2)

(57)







0

05

1

15

2

25

3

35

4

45

5


OSMG

Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


HOSM-IGC (120573 = 40)

HOSM-IGC (120573 = 50)

HOSM-IGC (120573 = 30)



6 Conclusions



119885becomes


175

180

185

190

195

200

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60 0 20 40X (km)


0

10

20

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus10

minus20



0

10

20

30

0 5 10 15 20Time (s)

OSMGHOSM-IGC

minus10

minus20

minus30

minus40

Actu

ator

defl

ectio

n (d

eg)


0

50

100

150

200

15 16 17 18Time (s)


Miss

ile ac

cele

ratio

n (G

)

minus50


Appendices



0= 1205821

= 1205822

= 1205823

= 50


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0

100

200

300

400

0 02 04Time (s)

h(t)

z3

h(t) and z3

minus100

minus200

minus300

minus400

minus5000 02 04

Time (s)

g(t)

z2

g(t) and z2

minus100

minus200

minus300

minus400

f(t)

z1

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

F(t)

z0

F(t) and z0

minus0005

minus001



= 01 1199111

= 1199112

= 1199113


0

= V0

V0

= minus120582011987114 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

34 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987113 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

23 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987112 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

12 sign (1199112

minus V1) + 1199113

3

= minus1205823119871 sign (119911

3minus V2)

(A1)


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0 02 04Time (s)

0

200

400

0 02 04Time (s)

h(t) and z3

minus200

minus400

minus600

minus800

g(t) and z2

minus100

minus200

minus300

minus400

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

minus16

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0



int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +








0 02 04

0

0005

001

0015

002

0025

003

0035

004

Time (s)0 02 04

0

01

02

03

04

05

Time (s)

0 02 04

0

05

1

15

2

Time (s)0 02 04

0

05

1

15

2

25

3

35

4

Time (s)


minus04

minus05

f(t) and z1

minus2

minus15

minus1

minus05

minus1

minus05

minus01

minus02

minus03

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0




10038161003816100381610038161003816120590(119894)

10038161003816100381610038161003816le 120583120591119903minus119894

119894 = 0 119903 minus 1 (A2)








2has an obviously






0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)


= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825





Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2


1063 Kgsdotm2



See Table 7



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












HOSM-IGC(120573 = 10)

119881119905= 2000ms 115m 086 073m

119881119905= 3000ms 261m 117 106

119881119905= 4000ms 542m 156 134

0

1

2

0 5 10 15 20Time (s)

72 74 76 78

0

02

minus02

minus04

Actu

ator

defl

ectio

n (d

eg)

minus1

minus2

minus3

minus4

HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)




119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

=

0 g 119905 lt 10 sec

5 g 119905 gt 10 sec

(56)






119881119872





15

17

19

21

23

25

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60X (km)







0

5

10

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus5

minus10


0

10

20

30

0 5 10 15 20Time (s)


tgo = 3 s (diverge)

tgo = 2 s (diverge)


minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 10)



0

10

20

30

0 5 10 15 20Time (s)


tgo = 1 s (diverge)

tgo = 2 s (diverge)


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

120573 = 30)HOSM-IGC (


0

10

20

30

0 5 10 15 20Time (s)

tgo = 1 s (diverge)




minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 50)






HOSM-IGC120573 = 30

HOSM-IGC120573 = 40

HOSM-IGC120573 = 50

OSMG

tgo = 1second 43539 3224 28936 36116

tgo = 2seconds 25424 18665 09322 35534

tgo = 3seconds 08124 08265 07538 11959

Time (s)0 5 10 15 20

0

10

20

30OSMG


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)









119881119872






119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 40 sin(120587119905

2)

(57)







0

05

1

15

2

25

3

35

4

45

5


OSMG

Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


HOSM-IGC (120573 = 40)

HOSM-IGC (120573 = 50)

HOSM-IGC (120573 = 30)



6 Conclusions



119885becomes


175

180

185

190

195

200

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60 0 20 40X (km)


0

10

20

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus10

minus20



0

10

20

30

0 5 10 15 20Time (s)

OSMGHOSM-IGC

minus10

minus20

minus30

minus40

Actu

ator

defl

ectio

n (d

eg)


0

50

100

150

200

15 16 17 18Time (s)


Miss

ile ac

cele

ratio

n (G

)

minus50


Appendices



0= 1205821

= 1205822

= 1205823

= 50


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0

100

200

300

400

0 02 04Time (s)

h(t)

z3

h(t) and z3

minus100

minus200

minus300

minus400

minus5000 02 04

Time (s)

g(t)

z2

g(t) and z2

minus100

minus200

minus300

minus400

f(t)

z1

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

F(t)

z0

F(t) and z0

minus0005

minus001



= 01 1199111

= 1199112

= 1199113


0

= V0

V0

= minus120582011987114 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

34 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987113 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

23 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987112 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

12 sign (1199112

minus V1) + 1199113

3

= minus1205823119871 sign (119911

3minus V2)

(A1)


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0 02 04Time (s)

0

200

400

0 02 04Time (s)

h(t) and z3

minus200

minus400

minus600

minus800

g(t) and z2

minus100

minus200

minus300

minus400

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

minus16

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0



int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +








0 02 04

0

0005

001

0015

002

0025

003

0035

004

Time (s)0 02 04

0

01

02

03

04

05

Time (s)

0 02 04

0

05

1

15

2

Time (s)0 02 04

0

05

1

15

2

25

3

35

4

Time (s)


minus04

minus05

f(t) and z1

minus2

minus15

minus1

minus05

minus1

minus05

minus01

minus02

minus03

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0




10038161003816100381610038161003816120590(119894)

10038161003816100381610038161003816le 120583120591119903minus119894

119894 = 0 119903 minus 1 (A2)








2has an obviously






0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)


= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825





Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2


1063 Kgsdotm2



See Table 7



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

5

10

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus5

minus10


0

10

20

30

0 5 10 15 20Time (s)


tgo = 3 s (diverge)

tgo = 2 s (diverge)


minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 10)



0

10

20

30

0 5 10 15 20Time (s)


tgo = 1 s (diverge)

tgo = 2 s (diverge)


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

120573 = 30)HOSM-IGC (


0

10

20

30

0 5 10 15 20Time (s)

tgo = 1 s (diverge)




minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)

HOSM-IGC (120573 = 50)






HOSM-IGC120573 = 30

HOSM-IGC120573 = 40

HOSM-IGC120573 = 50

OSMG

tgo = 1second 43539 3224 28936 36116

tgo = 2seconds 25424 18665 09322 35534

tgo = 3seconds 08124 08265 07538 11959

Time (s)0 5 10 15 20

0

10

20

30OSMG


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)









119881119872






119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 40 sin(120587119905

2)

(57)







0

05

1

15

2

25

3

35

4

45

5


OSMG

Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


HOSM-IGC (120573 = 40)

HOSM-IGC (120573 = 50)

HOSM-IGC (120573 = 30)



6 Conclusions



119885becomes


175

180

185

190

195

200

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60 0 20 40X (km)


0

10

20

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus10

minus20



0

10

20

30

0 5 10 15 20Time (s)

OSMGHOSM-IGC

minus10

minus20

minus30

minus40

Actu

ator

defl

ectio

n (d

eg)


0

50

100

150

200

15 16 17 18Time (s)


Miss

ile ac

cele

ratio

n (G

)

minus50


Appendices



0= 1205821

= 1205822

= 1205823

= 50


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0

100

200

300

400

0 02 04Time (s)

h(t)

z3

h(t) and z3

minus100

minus200

minus300

minus400

minus5000 02 04

Time (s)

g(t)

z2

g(t) and z2

minus100

minus200

minus300

minus400

f(t)

z1

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

F(t)

z0

F(t) and z0

minus0005

minus001



= 01 1199111

= 1199112

= 1199113


0

= V0

V0

= minus120582011987114 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

34 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987113 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

23 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987112 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

12 sign (1199112

minus V1) + 1199113

3

= minus1205823119871 sign (119911

3minus V2)

(A1)


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0 02 04Time (s)

0

200

400

0 02 04Time (s)

h(t) and z3

minus200

minus400

minus600

minus800

g(t) and z2

minus100

minus200

minus300

minus400

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

minus16

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0



int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +








0 02 04

0

0005

001

0015

002

0025

003

0035

004

Time (s)0 02 04

0

01

02

03

04

05

Time (s)

0 02 04

0

05

1

15

2

Time (s)0 02 04

0

05

1

15

2

25

3

35

4

Time (s)


minus04

minus05

f(t) and z1

minus2

minus15

minus1

minus05

minus1

minus05

minus01

minus02

minus03

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0




10038161003816100381610038161003816120590(119894)

10038161003816100381610038161003816le 120583120591119903minus119894

119894 = 0 119903 minus 1 (A2)








2has an obviously






0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)


= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825





Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2


1063 Kgsdotm2



See Table 7



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












HOSM-IGC120573 = 30

HOSM-IGC120573 = 40

HOSM-IGC120573 = 50

OSMG

tgo = 1second 43539 3224 28936 36116

tgo = 2seconds 25424 18665 09322 35534

tgo = 3seconds 08124 08265 07538 11959

Time (s)0 5 10 15 20

0

10

20

30OSMG


minus10

minus20

minus30

Miss

ile ac

cele

ratio

n (G

)









119881119872






119910119879

= 119881119879cos 120579119879

119879

= minus119881119879sin 120579119879

120579119879

=119886119879

119881119879

119886119879

= 40 sin(120587119905

2)

(57)







0

05

1

15

2

25

3

35

4

45

5


OSMG

Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


HOSM-IGC (120573 = 40)

HOSM-IGC (120573 = 50)

HOSM-IGC (120573 = 30)



6 Conclusions



119885becomes


175

180

185

190

195

200

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60 0 20 40X (km)


0

10

20

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus10

minus20



0

10

20

30

0 5 10 15 20Time (s)

OSMGHOSM-IGC

minus10

minus20

minus30

minus40

Actu

ator

defl

ectio

n (d

eg)


0

50

100

150

200

15 16 17 18Time (s)


Miss

ile ac

cele

ratio

n (G

)

minus50


Appendices



0= 1205821

= 1205822

= 1205823

= 50


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0

100

200

300

400

0 02 04Time (s)

h(t)

z3

h(t) and z3

minus100

minus200

minus300

minus400

minus5000 02 04

Time (s)

g(t)

z2

g(t) and z2

minus100

minus200

minus300

minus400

f(t)

z1

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

F(t)

z0

F(t) and z0

minus0005

minus001



= 01 1199111

= 1199112

= 1199113


0

= V0

V0

= minus120582011987114 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

34 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987113 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

23 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987112 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

12 sign (1199112

minus V1) + 1199113

3

= minus1205823119871 sign (119911

3minus V2)

(A1)


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0 02 04Time (s)

0

200

400

0 02 04Time (s)

h(t) and z3

minus200

minus400

minus600

minus800

g(t) and z2

minus100

minus200

minus300

minus400

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

minus16

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0



int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +








0 02 04

0

0005

001

0015

002

0025

003

0035

004

Time (s)0 02 04

0

01

02

03

04

05

Time (s)

0 02 04

0

05

1

15

2

Time (s)0 02 04

0

05

1

15

2

25

3

35

4

Time (s)


minus04

minus05

f(t) and z1

minus2

minus15

minus1

minus05

minus1

minus05

minus01

minus02

minus03

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0




10038161003816100381610038161003816120590(119894)

10038161003816100381610038161003816le 120583120591119903minus119894

119894 = 0 119903 minus 1 (A2)








2has an obviously






0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)


= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825





Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2


1063 Kgsdotm2



See Table 7



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

05

1

15

2

25

3

35

4

45

5


OSMG

Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)

0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


0

05

1

15

2

25

3

35

4

45

5


Miss

(m)


HOSM-IGC (120573 = 40)

HOSM-IGC (120573 = 50)

HOSM-IGC (120573 = 30)



6 Conclusions



119885becomes


175

180

185

190

195

200

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60 0 20 40X (km)


0

10

20

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus10

minus20



0

10

20

30

0 5 10 15 20Time (s)

OSMGHOSM-IGC

minus10

minus20

minus30

minus40

Actu

ator

defl

ectio

n (d

eg)


0

50

100

150

200

15 16 17 18Time (s)


Miss

ile ac

cele

ratio

n (G

)

minus50


Appendices



0= 1205821

= 1205822

= 1205823

= 50


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0

100

200

300

400

0 02 04Time (s)

h(t)

z3

h(t) and z3

minus100

minus200

minus300

minus400

minus5000 02 04

Time (s)

g(t)

z2

g(t) and z2

minus100

minus200

minus300

minus400

f(t)

z1

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

F(t)

z0

F(t) and z0

minus0005

minus001



= 01 1199111

= 1199112

= 1199113


0

= V0

V0

= minus120582011987114 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

34 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987113 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

23 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987112 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

12 sign (1199112

minus V1) + 1199113

3

= minus1205823119871 sign (119911

3minus V2)

(A1)


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0 02 04Time (s)

0

200

400

0 02 04Time (s)

h(t) and z3

minus200

minus400

minus600

minus800

g(t) and z2

minus100

minus200

minus300

minus400

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

minus16

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0



int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +








0 02 04

0

0005

001

0015

002

0025

003

0035

004

Time (s)0 02 04

0

01

02

03

04

05

Time (s)

0 02 04

0

05

1

15

2

Time (s)0 02 04

0

05

1

15

2

25

3

35

4

Time (s)


minus04

minus05

f(t) and z1

minus2

minus15

minus1

minus05

minus1

minus05

minus01

minus02

minus03

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0




10038161003816100381610038161003816120590(119894)

10038161003816100381610038161003816le 120583120591119903minus119894

119894 = 0 119903 minus 1 (A2)








2has an obviously






0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)


= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825





Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2


1063 Kgsdotm2



See Table 7



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










175

180

185

190

195

200

TargetHOSM-IGCOSMG

Y(k

m)

0 20 40 60 0 20 40X (km)


0

10

20

0 5 10 15 20Time (s)

TargetOSMGHOSM-IGC

Miss

ile an

d ta

rget

acce

lera

tion

(G)

minus10

minus20



0

10

20

30

0 5 10 15 20Time (s)

OSMGHOSM-IGC

minus10

minus20

minus30

minus40

Actu

ator

defl

ectio

n (d

eg)


0

50

100

150

200

15 16 17 18Time (s)


Miss

ile ac

cele

ratio

n (G

)

minus50


Appendices



0= 1205821

= 1205822

= 1205823

= 50


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0

100

200

300

400

0 02 04Time (s)

h(t)

z3

h(t) and z3

minus100

minus200

minus300

minus400

minus5000 02 04

Time (s)

g(t)

z2

g(t) and z2

minus100

minus200

minus300

minus400

f(t)

z1

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

F(t)

z0

F(t) and z0

minus0005

minus001



= 01 1199111

= 1199112

= 1199113


0

= V0

V0

= minus120582011987114 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

34 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987113 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

23 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987112 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

12 sign (1199112

minus V1) + 1199113

3

= minus1205823119871 sign (119911

3minus V2)

(A1)


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0 02 04Time (s)

0

200

400

0 02 04Time (s)

h(t) and z3

minus200

minus400

minus600

minus800

g(t) and z2

minus100

minus200

minus300

minus400

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

minus16

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0



int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +








0 02 04

0

0005

001

0015

002

0025

003

0035

004

Time (s)0 02 04

0

01

02

03

04

05

Time (s)

0 02 04

0

05

1

15

2

Time (s)0 02 04

0

05

1

15

2

25

3

35

4

Time (s)


minus04

minus05

f(t) and z1

minus2

minus15

minus1

minus05

minus1

minus05

minus01

minus02

minus03

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0




10038161003816100381610038161003816120590(119894)

10038161003816100381610038161003816le 120583120591119903minus119894

119894 = 0 119903 minus 1 (A2)








2has an obviously






0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)


= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825





Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2


1063 Kgsdotm2



See Table 7



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0

100

200

300

400

0 02 04Time (s)

h(t)

z3

h(t) and z3

minus100

minus200

minus300

minus400

minus5000 02 04

Time (s)

g(t)

z2

g(t) and z2

minus100

minus200

minus300

minus400

f(t)

z1

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

F(t)

z0

F(t) and z0

minus0005

minus001



= 01 1199111

= 1199112

= 1199113


0

= V0

V0

= minus120582011987114 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

34 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987113 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

23 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987112 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

12 sign (1199112

minus V1) + 1199113

3

= minus1205823119871 sign (119911

3minus V2)

(A1)


0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0 02 04Time (s)

0

200

400

0 02 04Time (s)

h(t) and z3

minus200

minus400

minus600

minus800

g(t) and z2

minus100

minus200

minus300

minus400

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

minus16

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0



int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +








0 02 04

0

0005

001

0015

002

0025

003

0035

004

Time (s)0 02 04

0

01

02

03

04

05

Time (s)

0 02 04

0

05

1

15

2

Time (s)0 02 04

0

05

1

15

2

25

3

35

4

Time (s)


minus04

minus05

f(t) and z1

minus2

minus15

minus1

minus05

minus1

minus05

minus01

minus02

minus03

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0




10038161003816100381610038161003816120590(119894)

10038161003816100381610038161003816le 120583120591119903minus119894

119894 = 0 119903 minus 1 (A2)








2has an obviously






0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)


= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825





Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2


1063 Kgsdotm2



See Table 7



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

0005

001

0015

002

0025

003

0035

004

0 02 04Time (s)

0

2

0 02 04Time (s)

0

100

200

300

0 02 04Time (s)

0

200

400

0 02 04Time (s)

h(t) and z3

minus200

minus400

minus600

minus800

g(t) and z2

minus100

minus200

minus300

minus400

f(t) and z1

minus2

minus4

minus6

minus8

minus10

minus12

minus14

minus16

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0



int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +








0 02 04

0

0005

001

0015

002

0025

003

0035

004

Time (s)0 02 04

0

01

02

03

04

05

Time (s)

0 02 04

0

05

1

15

2

Time (s)0 02 04

0

05

1

15

2

25

3

35

4

Time (s)


minus04

minus05

f(t) and z1

minus2

minus15

minus1

minus05

minus1

minus05

minus01

minus02

minus03

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0




10038161003816100381610038161003816120590(119894)

10038161003816100381610038161003816le 120583120591119903minus119894

119894 = 0 119903 minus 1 (A2)








2has an obviously






0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)


= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825





Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2


1063 Kgsdotm2



See Table 7



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04

0

0005

001

0015

002

0025

003

0035

004

Time (s)0 02 04

0

01

02

03

04

05

Time (s)

0 02 04

0

05

1

15

2

Time (s)0 02 04

0

05

1

15

2

25

3

35

4

Time (s)


minus04

minus05

f(t) and z1

minus2

minus15

minus1

minus05

minus1

minus05

minus01

minus02

minus03

F(t) and z0

minus0005

minus001

h(t)

z3g(t)

z2

f(t)

z1F(t)

z0




10038161003816100381610038161003816120590(119894)

10038161003816100381610038161003816le 120583120591119903minus119894

119894 = 0 119903 minus 1 (A2)








2has an obviously






0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)


= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825





Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2


1063 Kgsdotm2



See Table 7



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














2has an obviously






0

= V0

V0

= minus120582011987116 10038161003816100381610038161199110 minus

1199021003816100381610038161003816

56 sign (1199110

minus 119902) + 1199111

1

= V1

V1

= minus120582111987115 10038161003816100381610038161199111 minus V

0

1003816100381610038161003816

45 sign (1199111

minus V0) + 1199112

2

= V2

V2

= minus120582211987114 10038161003816100381610038161199112 minus V

1

1003816100381610038161003816

34 sign (1199112

minus V1) + 1199113

3

= V3

V3

= minus120582311987113 10038161003816100381610038161199113 minus V

2

1003816100381610038161003816

23 sign (1199113

minus V2) + 1199114

4

= V4

V4

= minus120582411987112 10038161003816100381610038161199114 minus V

3

1003816100381610038161003816

12 sign (1199114

minus V3) + 1199115

5

= V5

V5

= minus1205825119871 sign (119911

5minus V4)

(B1)


= 01 1205820

= 1205821

= 1205822

= 1205823

= 1205824

=

1205825





Table 7

Length 365m119871 ref 365m119883119866

177m119878ref 0026m2


1063 Kgsdotm2



See Table 7



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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